Mumbai had set a modest target of 140 in their 20 overs, but Chennai repeatedly threw wickets away to post a total of 79, the lowest in the IPL this season.
Chennai’s highest scorer with the bat was opener Mike Hussey (22), with Ravindra Jadeja (20) and MS Dhoni (10) the only others to reach double figures as three went for ducks.
There was a bizarre incident too, as Mumba's Kieron Pollard managed to drop three catches in a row as Hussey repeatedly cut to the West Indies bowler off Mitchell Johnson's bowling.
In the later match, Rajasthan beat Pune's total of 178 for 4 with one ball to spare, as Stuart Binny smacked a four to clinch victory.
Ajinkya Rahane was the hero, his 67 setting the Royals on their way.
"180 was not easy to chase," he said. "But overall everyone did well. Especially our bowlers came back, and I think they were 10-15 short.
"We just wanted to play cricketing shots, our normal game. I was disappointed to get when I did, but Stuart and Sanju (Samson) did really well."