The game was turned decisively in the hosts' favour early in the second half as Bolton had Ricardo Gardner sent off after collecting a second booking and Joe Allen gave Swansea the lead from the resulting free-kick.
Scott Sinclair got the second as he converted his third spot-kick of the season in the 57th minute after Darren Pratley, formerly of Swansea, had fouled Angel Rangel in the area.
Bolton were given a lifeline when Swansea conceded at home for the first time this term as Graham put through his own net, but the club record signing redeemed himself by slotting his fourth goal in four games in added time as Swansea claimed a vital three points ahead of upcoming fixtures against Liverpool and Manchester United.
Bolton had defended obdurately during the opening half, and came close to taking a first-half lead through David Ngog, but this defeat will only increase the pressure on manager Owen Coyle with his side ensconced in the relegation zone.
Swansea had made a typically fluent opening but became frustrated by a well-organised visiting defence as the half wore on.
They came close to an opener when Sinclair just failed to get a touch on Nathan Dyer's inviting ball across the box, while Mark Gower sliced a volley wide following assured build-up between Neil Taylor, Sinclair and Allen.
Gardner had dragged an early effort well wide at the other end after Dyer had been robbed by Mark Davies.
And it was Bolton who had the best effort of the opening 45 minutes as Ngog's smart turn and volley forced Michel Vorm to acrobatically tip the ball over the bar.
But Swansea received a huge boost within four minutes of the restart as they took the lead moments after Bolton were reduced to 10 men.
Graham had come within inches of finding the opener as he shot wide from a Sinclair pass before the game was turned on its head.
Gardner was dismissed after he picked up a second booking for a cynical tug on Dyer, and from the resulting free-kick Allen was afforded far too much room by Gary Cahill and Gretar Steinsson to shoot low into Jaaskelainen's bottom right-hand corner.
Dyer almost added a second after 50 minutes as he jinked his way into the box to force Jaaskelainen to block with his legs.
Coyle withdrew Martin Petrov to bring on Paul Robinson as a defensive reinforcement but it made little difference as Jaaskelainen was again called into action to parry an angled Sinclair drive behind.
But the resulting corner saw Pratley bring down Rangel and concede a penalty, which Sinclair planted into the right-hand corner for a 2-0 lead after 57 minutes.
Sinclair then shot just wide of the post with Bolton in disarray before Graham headed onto the bar with Jaaskelainen rooted to the spot.
But Wanderers were then given an unlikely avenue back into the game as Graham's attempt to block a Chris Eagles shot only succeeded in deflecting the ball beyond Vorm and into the net.
Sinclair had a great chance to seal the win with eight minutes left but could only hit the bar after being picked out by Dyer's intelligent cutback.
Coyle threw skipper Cahill up front for the final minutes but he had little impact and Graham wrapped up the win in injury time as he was given the freedom of the Bolton half to race clear and beat Jaaskelainen.