The Swedish centre-back scored an 89th minute goal to secure the points for the visitors.
In a match in which the two sides cancelled each other out for long periods, Stuart Dallas fired Leeds in front on 74 minutes with a strike from the corner of the box.
Rovers responded well, getting back on level terms just 9 minutes later thanks to another long-range strike from Elliott Bennett
But whilst a draw would have been a fair result, it was the side in yellow who claimed all three points when Jansson struck to break Rovers hearts.
Owen Coyle made two changes from the side that defeated Blackpool in the FA Cup at the weekend as Charlie Mulgrew and Liam Feeney returned to the side. Hope Akpan missed out through injury, whilst Craig Conway dropped to the subs bench along side new signing Lucas Joao.
Rovers had the ball in the back of the net after just four minutes when Sam Gallagher bundled it home. But referee Geoff Eltringham ruled that the striker had used a little too much force against Leeds keeper Rob Green in attempting to get on the end of Bennett's centre.
But despite that bright start, the game became somewhat of a midfield battle with neither side giving up much in the way of goalmouth openings.
A curling free-kick from Chris Wood flashed just wide of the far post, whilst a Luke Ayling centre was just too high for Kemar Roofe.
Rovers started the second half the stronger of the two sides with a flurry of corner kicks, whilst a dangerous centre from Feeney was put over his own bar by Luke Ayling.
As the clock ticked on the game started to become stretched with space opening up to be exploited. A neat interchange of passing between Mulgrew, Bennett and Danny Graham saw play open up for Gallagher, but from distance he fired well wide.
Then after Bennett was fouled to the left of the area, his free-kick seemed to be heading towards Mulgrew at the back post, but Green just got there in the nick of time to touch the ball away.
But in an instant play was shifted down the other end for Leeds to grab an unlikely opener. Liam Bridcutt hit a diagonal ball over the Rovers back line to where Dallas cut inside and finished well into the opposite corner of the goal.
Gallagher saw a shot deflect wide as Rovers looked for a swift response, whilst a pair of corner saw goalmouth scrambles saw Leeds just about survived.
But with seven minutes remaining Rovers got the goal that their pressure deserved. Keeping possession well on the left, the ball was shifted inside to Bennett and he fired home from 25-yards.
It wasn't quite a repeat of the stunning strike from the weekend that found the top corner, this time his low strike crept home at the far post.
Sensing the chance to turn the match around, Rovers pressed. Sub Lucas Joao almost marked his debut with a goal when he latched onto a pass from Graham, but he scooped his effort over the bar.
That miss would prove crucial as Leeds went up the other end to earn themselves a corner. Dallas delivered from the left and Jansson arrived unmarked to head into the back of the net to clinch an undeserved victory.
ROVERS: Steele, Nyambe, Greer, Lenihan, Williams, Feeney (sub Conway 78 mins), Lowe (c) (sub Joao 78 mins), Mulgrew, Bennett, Gallagher, Graham.
Subs not used: Raya (GK), Guthrie, Tomlinson, Mahoney, Brown.
Goals: Bennett 83 mins.
LEEDS: Green, Ayling, Bartley, Roofe (sub Doukara 61 mins), Wood, Jansson, O'Kane (sub Vieira 78 mins), Dallas, Sacko (sub Hernandez 46 mins), Bridcutt (c), Berardi.
Subs not used: Silvestri (GK), Antonsson, Phillips, Coyle.
Goals: Dallas 74 mins, Jansson 89 mins
Booked: Jansson 35 mins.
Attendance: 17,026 (including 6402 visiting supporters)