Southampton looked set to end a week of turmoil, which included the resignation of chairman Nicola Cortese on Wednesday, with victory after first-half goals from Jay Rodriguez and Dejan Lovren.
Yet the visitors' recent resurgence continued as Borini halved the deficit before half-time, and Johnson maintained his excellent form with a superb right-footed strike to level an exhilarating contest.
Southampton dominated possession early on and deservedly went ahead through Rodriguez's fierce half-volley, before Lovren volleyed home Steven Davis' corner at the near post on 31 minutes.
However, Sunderland reduced their arrears almost immediately when Borini calmly slotted home.
The visitors had ample opportunities to pull clear after the break, but were made to rue their wastefulness when Johnson fired home at the near post.
The result sees Sunderland move up one spot to 18th, while Southampton remain ninth.
Sunderland made two changes with Sebastian Larsson and Jozy Altidore replacing Jack Colback and Steven Fletcher, while the visitors brought in Calum Chambers for Nathaniel Clyne.
Southampton's early dominance was rewarded after four minutes when the hosts failed to clear their lines, allowing Rodriguez time to bring the ball out of the air and rifle home a superb half-volley from 25 yards.
Buoyed by the opener, Jack Cork almost capitalised on Vito Mannone's poor clearance by lofting an effort onto the roof of the net.
The home fans were becoming restless and their mood was not helped when Rickie Lambert dragged wide from Morgan Schneiderlin's neat flick after being put through one-on-one with Mannone.
Wes Brown was caught in possession soon after by Lambert, who laid into the path of Rodriguez only for the forward to shoot tamely at Mannone.
Southampton's second goal deservedly arrived 31 minutes, Lovren escaping the attention of Brown to volley home Davis' right-wing corner.
Within the space of a minute Sunderland clawed their way back into the game when Borini collected Johnson's long ball and guided his shot past Artur Boruc.
The game opened up after the break and Southampton should have scored again when good work by Adam Lallana led to Rodriguez firing straight at Mannone from the edge of the box on 56 minutes.
The away side continued to press with Lovren seeing a free header saved, before Rodriguez met Lambert's centre from the right on the hour mark but failed to hit the target.
Southampton were almost punished when Borini sprinted away from the defence and slotted under Boruc, only to denied by the linesman's flag.
However, the home fans were sent into delirium on 71 minutes when Ki Sung-Yueng charged forward and found Craig Gardner. His pass picked out Johnson on the right-hand side of the box and the winger's thunderous strike squirmed in at the near post.
With the momentum behind them Sunderland pushed for the winner, and Johnson almost delivered for Gus Poyet's men with a half-volley that was smartly stopped by Boruc.
Southampton's day worsened in the closing stages when Gaston Ramirez was carried off on a stretcher with an apparent ankle injury following a challenge by Brown.
