Sent into bat, Punjab were fairly well placed at 106 for three at the start of 15th over, when top-scorer David Hussey's dismissal for a 36-ball 41 triggered a collapse from which the hosts could never recover.
Some mindless batting by the hosts only made the task easier for Chennai, who showed discipline on the field.
Punjab relied on Hussey's knock and his 56-run stand for the fourth wicket with Gurkeerat Singh after suffering three early blows at the PCA Stadium.
Adam Gilchrist found the fence twice in the next over, an outside edge to the third man boundary followed by a cover drive that was a connoisseur's delight.
But Punjab could not sustain the tempo for long, as Nannes gave Chennai their first breakthrough in the form of Gilchrist in the third over.
Even though he was talking to the TV commentators, Hussey was focussed enough to dive to his left and get hold of Gilchrist's whip off the pads.
Ravindra Jadeja had Manan Vohra after the batsman helped himself to three boundaries for an 11-ball 16.
There was a 56-run partnership for the fourth wicket between David Hussey and Gurkeerat Singh, before the Aussie was snared by off-spinner Ravichandran Aswhin.
Gurkeerat followed after the addition of 10 runs, falling prey to Bravo.
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