Rovers’ second loss in the middle-eights leaves them facing a battle with Salford Red Devils and Huddersfield Giants to hang on to their Super League place for 2017.
Leigh, who have topped the Championship for three successive seasons, could confirm promotion if they win at home to Huddersfield next weekend.
Even if they lose that, a win against part-timers Batley Bulldogs the following week would send them up.
Rovers led 12-0 early on, but were sunk by three tries in five minutes early in the second half as the visitors maintained their 100 per cent record.
Hull KR were without Kelly who was suspended by Rovers over a “breach of club discipline”.
In a statement issued before the match, Rovers said Kelly was granted permission to return to his native Australia for family reasons two weeks ago, but arrived back late and missed “important preparation” for the tussle with Leigh.
Club head of rugby Jamie Peacock described Kelly’s actions as “very disappointing” and said Rovers will be speaking to the player and his agent over the coming week to “reinforce our expectations”.
Rovers made light of the pre-game disruption in the early stages as they raced into a big lead, but Leigh weathered the storm to trail by only six points at the break.
The Championship side were placed on a team warning inside the opening four minutes after conceding three penalties, the last of which Josh Mantellato kicked to edge Rovers ahead.
Ken Sio crossed two minutes later and Mantellato also touched down before booting another penalty to make it 12-0 after 16.
Leigh were placed on a second warning, but made the most of a penalty their way on 25 minutes when Sam Hopkins barged over from close range and Martyn Ridyard tagged on the conversion.
It stayed 12-6 until seven minutes in to the second half when Andrew Dixon went over from a pass by Ridyard, whose conversion levelled the scores.
Three minutes later Josh Drinkwater put up a hig kick, Ben Cockayne lost it and Matty Dawson touched down.
Video referee James Child confirmed the try. There was no goal, but Leigh were over again in the next set, Drinkwater backing up Cory Paterson’ break.
Ridyard’s conversion opened a 10-point gap. Rovers got a foothold back in the game with 15 minutes remaining when Cockayne carved out a try for Matty Marsh and Mantellato added the extras.
Ridyard landed a long-range penalty with 10 minutes left and Drinkwater failed with two drop goals before Ridyard landed a one-pointer to complete the scoring.
Rovers’ defeat was more delight for the black and white half of the city, a week after Hull’s Challenge Cup triumph.
Hull KR were taunted during the first half by a light aircraft which circled the ground towing a banner reading “Hull FC Wembley winners 2016”.