The game was end-to-end until the dramatic closing stages, with relegation-threatened Aycliffe looking as if they would hold on.
Bishops keeper Peter Jeffries pulled off two excellent saves, while at the other end, Chris Lawson and Steve Tobin both hit the woodwork for Bishops.
Aycliffe took the lead on 61 minutes from a Danny Earl penalty after Steve Salvin was adjudged to have fouled Josh Knox.
Aycliffe looked as if they would hold on for three valuable points until the 85th minute when Dan Hawkins made a good run down the left and crossed low for Johnson to stab the ball past Scott Pocklington.
Three minutes later, Bishops took the lead from another dubious penalty, this time awarded for handball by an Aycliffe defender in a goalmouth scramble. Johnson’s spot-kick was blocked by Pocklington’s left hand, but the striker reacted first to turn the ball home.
And in stoppage time he completed his hat-trick when he ran 40 yards and slotted the ball past the advancing keeper.
Bishop Auckland manager Colin Myers said: "I didn't think either of them were penalties. I thought theirs was a clean and fair tackle, and the second the ball hit somebody on the back of the head."
Spennymoor Town remained seven points behind leaders Darlington 1883 with a 2-0 derby win over neighbours Shildon at the Brewery Field.
Shildon keeper Keith Finch pulled off two good saves from Micky Rae and Gavin Cogdon before Moors took the lead on 22 minutes, when Finch could only a parry a fierce low shot by Rae into the path of Cogdon, who tapped in his fourteenth league goal of the season.
Shildon slowly got back into the game, and missed a great chance to get back into the game just after half-time when Robbie Briggs was fouled by Micky Laws just inside the area, but Moors keeper Robbie Deane got down well to his right to save Chris Hughes’ spot kick.
Moors made sure of three valuable points in the last minute when Joe Walton set up Mark Davison to score from a few yards.
Shildon manager Gary Forrest said: "I thought we were a little unfortunate, the missed penalty was probably the biggest factor. I genuinely believe that had we scored then we would have gone on to win the game as we had the momentum. You can't go to places like Spennymoor and miss the chances we did today as you don't get that many."
Spennymoor manager Jason Ainsley said: "It was a scrappy game on a difficult surface. I felt the penalty save was the difference even though it was a yard outside the area. We are going through a spell in which were not playing as fluid as earlier in the season, but what's pleasing is that we are still winning the games and are strong defensively."
Consett hadn’t received a red card this season until today, when they had two sent-off in the 2-0 home defeat by Durham City, who are now seventh from bottom.
Micheal Pearson was dismissed just after half-time for a professional foul, and on 68 minutes, Anthony Lund was sent-off for deliberate handball and Olly Hotchkiss scored from the spot.
Three minutes later, Durham scored their second when Scott Fenwick fired home from the edge of the box.
Consett manager Kenny Lindoe said: "The two sendings-off early in the second half by a young and very poor ref killed the game. Their first goal was a penalty for a handball, but only after our keeper was fouled. The second came from a corner that was very clearly a goal kick."
Synthonia won the Billingham derby against Town by 3-1. Danny McWilliams and Dan Johnson combined to set up Nathan Wilson to put them into the lead, and on 19 minutes Johnson made it 2-0 with a low shot.
Micky Dunwell pulled a goal back with a header from a Harrison Davies cross, but Wilson made it 3-1 on 33 minutes from the penalty spot after he was fouled by Richie Gaston.
Synners` assistant-manager Lee Tucker said: "It capped off a good Christmas period for us with two excellent wins. Without being at our best we fully deserved our victory. We scored three good goals playing some brilliant football at times."
Town manager Carl Jarrett said: "My lads never turned up today. They were totally outplayed by a good footballing team."
South Shields are still rock bottom after they lost 3-2 at Sunderland RCA.
Nick Kane and recent signing Gary Shaw put RCA 2-0 up, but Peter Ramsey and Jonny Wightman fought back for Shields. However, Nathan Monson popped up with the winner for RCA.
Sunderland RCA manager Neil Hixon said: "By half-time we should have been winning five or six. The second half was very tight, Shields probably edged it."
Bedlington Terriers came back from 3-1 down to earn a point in a 3-3 draw at Ashington, Phil Bannister and Mark Doninger scoring late goals.
Andy Johnson (2) and James Taylor scored for Ashington, with Bannister scoring Bedlington’s other goal.
Ashington assistant-manager Perry Briggs said: "We threw the game away. We should have been five or six before they got their second. We've got to be more ruthless in both boxes."
In the Second Division, fifth-placed Jarrow Roofing hit double figures against second-bottom Horden CW, thrashing them 10-2.
Andy Appleby became the third player from the division to score five in a game this season, with Darren Palmer, David Carson, Liam McBryde, Dan Smart and Anthony Myers getting the others. Peter Crawford scored both for Horden in the first half.
Horden manager Peter Mulcaster said: "There have to be changes following the shambolic performance. Too many players didn't want to know, too many with a Sunday league mentality. Although we were playing a team going for promotion and were short of six players, the manner in which we were beaten cannot be accepted. It's been a long time since I was embarrassed with a team."
Jarrow Roofing manager Richie McLoughlin said: "The lads put in a lot of hard work today and fully deserved the win."
Leaders Ryhope CW were held to a 2-2 draw at home by Seaham Red Star.
Channon North rounded the Ryhope keeper to put them in the lead, then Paul Kane made it 2-0 from the spot following a foul by Greg Swansbury.
Seaham manager Mark Collingwood said: "We were fantastic and should have won. We needed a third goal at 2-1 but we missed chances. Credit to Ryhope for the way they dug in and scored late on. We would have taken a point before the game."
Chris Trewick pulled a goal back in a scramble just before half-time, and after Seaham hit the post, Ryhope striker Johnny Butler levelled with ten minutes left, with his 33rd league goal of the season.
Crook Town are now level on points with Ryhope after winning 3-2 at Tow Law Town.
Mattie Scott gave Lawyers the lead , but Warren Byrne and Craig Price hit back for Crook, Adam McVickers levelled for Lawyers, but Kyle Davis fired the winner for Crook with ten minutes left.
Tow Law manager Gary Anderson said: "Yet again another defeat at home in a game we deserved something from."
Northallerton Town and Darlington RA drew 1-1 at Ainderby Road.
The RA took the lead through Martin Butterworth after Steve Alison’s shot hit the bar, but Adam Emson levelled from the edge of the box with his nineteenth league goal of the season.
Northallerton manager Mark Fanning said: "Our first half performance was the worst 45 minutes of the season, and the RA could have been out of sight. We improved in the second half and got the equaliser but we hardly deserved it."
Brandon United scored in the last minute to earn a point against Esh Winning.
Dean Thexton twice put Esh into the lead, but first Owen Lancaster and Glenn Hendrix – right at the end – equalised.
Chester-le-Street beat Birtley Town 2-0, with goals by Mark Patterson and Michael Pattison.
Chester manager Colin Wake said: "Not a fluent performance by any stretch but had to work on a really heavy surface which cut up bad. Two goals from two subs and I feel a fully deserved three points."
Birtley manager Scott Oliver said: "We shaded the first half, and the longer the game went on, it was a free kick or a corner that would settle it. We didn't do enough in the last third."
The match between Thornaby and Stokesley SC was abandoned after 70 minutes because of an injury to Stokesley’s Matthew Robinson, with Thornaby winning 4-3 after trailing 3-0 at one point.
The games at Marske, Newcastle Benfield, Morpeth, Celtic Nation, North Shields and Whitehaven were all postponed.
Credit: Ray Simpson