Midfielder Andrich was taken off with a foot fracture while his teammate picked up a muscle injury. They will miss the last three league matches as well as their Europa League return leg next week.
"Leverkusen will be without Odilon Kossounou and Robert Andrich for the remainder of the season," the club said in a statement.
"Kossounou suffered a muscle injury in his right thigh. Andrich suffered a metatarsal fracture towards the end of the game."
Leverkusen, bidding for their first European final in 21 years, will have little time to recover before next Thursday's return leg against Rom, with VfB Stuttgart awaiting in the Bundesliga on Sunday.
Coach Xabi Alonso's team were on a 14-game unbeaten run in all competitions until last week's league loss to Cologne and they are battling to hold on to sixth place in the Bundesliga, which would secure a spot in next season's Europa League.
Leverkusen are on 48 points with three games left. VfL Wolfsburg are in seventh on 46 while Mainz 05 are on 45 in eighth spot.
Stuttgart, in 16th, are in the relegation playoff spot, level on points with 17th-placed VfL Bochum.
"There is still everything to fight for against Roma but our focus now is on Sunday and Stuttgart," Alonso told a press conference on Friday.
"It is a very important game for us and for Stuttgart. We have to recover quickly. We need a good game. We want to hold on to sixth place."
