The Rishabh Pant-led Delhi are making a late dash to enter the play-offs as they jump to fourth in the 10-team table led by Gujarat Titans who have already booked a top-two finish.
Delhi got off to a horror start after being put into bat first when David Warner fell first ball to Livingstone's off spin, giving away catch to backward point.
Fellow opener Sarfaraz Khan hit back with a 32-ball 16 before falling to left-arm medium-pace bowler Arshdeep Singh who ended with three wickets.
Marsh, who made a match-winning 89 in Delhi's previous match, stood firm as Livingstone rattled the middle order with key wickets of skipper Rishabh Pant (7) and Rovman Powell (2).
Marsh attacked, batted with composure when wickets fell and then accelerated in his knock laced with three sixes to boost Delhi's total.
In reply, Jonny Baristow smashed a 15-ball 28 but his departure and twin strikes in the sixth over by Thakur to send back Bhanuka Rajapaksa and Shikhar Dhawan hurt Punjab's chase.
Slow bowlers soon took over with left-arm orthodox Axar Patel and left-arm wrist spinner Kuldeep Yadav getting the big wickets of skipper Mayank Agarwal, for nought, and Livingstone, for three.
Axar and Kuldeep took two wickets each.
Punjab slipped to 82-7 before wicketkeeper-batsman Jitesh Sharma gave Delhi a scare with his 34-ball 44 but Thakur got him out with Warner taking a good low catch at long-off.