Mumbai were left ninth in the table, with Lucknow rooted to the bottom of the 10-team standings. Both sides have six points from 11 matches. The top four teams in the unified points table qualify for the playoffs.
Reigning champions Bengaluru climbed to the top of the standings with 14 points, while five-time winners Chennai are fifth with 12 after recovering from a sluggish start to their campaign.
Bengaluru won the toss and opted to bowl first in Raipur, where Mumbai posted 166-7.
India pace spearhead Jasprit Bumrah kept Mumbai in the contest with a superb penultimate over, conceding just three runs.
Bengaluru, however, held their nerve, chasing down 15 in the final over and sealing a two-wicket victory on the last ball to snap a two-match losing streak.
Bengaluru's top order - Jacob Bethell (27), Virat Kohli (0), Devdutt Padikkal (12) and captain Rajat Patidar (8) - failed to build momentum, but all-rounder Krunal Pandya anchored the chase with a composed 73 off 46 balls.
"I was cramping a lot - it started in my calf, then went to my glutes and back, so I was really struggling. I don't usually bat that long," Pandya, who is the elder brother of Mumbai captain Hardik Pandya and a former Mumbai player, said.
"I love tough situations and always prepare for them. I wanted to finish it myself, but couldn't, though the shot Bhuvi (Bhuvneshwar Kumar) played was the shot of the match," he added, referring to the six in the final over by the bowler.
Earlier, Chennai beat Lucknow by five wickets after Lucknow posted 203-8, thanks to Urvil Patel's blistering 23-ball 65 - the joint-fastest fifty in IPL history.