At the Wanders Stadium, Bangalore captain Kumble, 39, produced a superb spell of one for 13 in four overs, putting a brake on what had been a fast Lions scoring rate and dismissing Lions danger man Alviro Petersen.
Kohli, 21, made an audacious 49 not out off 29 balls to see the Indian Premier League team home on net run rate after they finished tied on points with the Lions and Mumbai Indians.
The Lions dominated the early part of the match. They raced to 57 for two in the first six overs, with captain Petersen and Vaughn van Jaarsveld both in superb form before Kumble came on to bowl.
Van Jaarsveld was unlucky to be run out for 24 in the eighth over, slipping in mid-pitch. Petersen went on to score 45 off 29 balls before he swung across the line and was bowled by the veteran Indian leg-spinner.
On a good batting pitch, the Lions were probably 20 runs short of a total that could be defended by an inexperienced bowling line-up.
Manish Pandey and Rahul Dravid got Bangalore off to a solid start, putting on 53 in eight overs but the Challengers also had a mid- innings stutter before Kohli's clean hitting proved decisive.