Marsh smashed a 48-ball 81 containing five sixes as LSG piled up 3-238 in their 20 overs.
The Australian was well supported by South African Aiden Markram (47 off 28) in a 99-run opening partnership, then West Indies' Nicholas Pooran (87no off 36), with whom he added 71 in five overs before holing out at deep point.
With 265 runs at 53 Marsh briefly became the tournament's highest scorer, but did not get to don the orange cap as he was overtaken in the same innings by Pooran.
The innings of the day came, however, in Tuesday's second match when Priyansh Arya punished Chennai Super KIngs for spurning four chances to catch him by battering an astonishing 39-ball century featuring nine sixes, a innings all the more remarkable for the rest of the top six making 23 between them.
That included Marcus Stoinis (4) and Glenn Maxwell (brilliantly caught and bowled for one by Ravi Ashwin), before Shashank Singh (52 off 36) provided support. That enabled Ricky Ponting's Punjab Kings, at one stage 5-85 to reach 6-219, Arya making 103 of them.
That was too much for CSK, even with the legendary MS Dhoni coming to the crease - to huge excitement even at the away venue of New Chandigarh - with CSK needing 68 off 25 balls.
Dhoni survived an umpire's call lbw and a run out chance to Stoinis to hit successive sixes off Lockie Ferguson - in between which Devon Conway was retired for 69 off 49 balls. Dhoni hit another six but was then caught for 27 off 12 and CSK's hopes went with him, the innings closing on 5-201.
Earlier at Eden Gardens Spencer Johnson was one of the bowlers hammered by Marsh, his three overs going for 46 without reward.
Yet Kolkata were on the verge of overhauling the huge target after captain Ajinkya Rahane blazed 61 off 35 balls and Venkatesh Iyer 45 off 29.
There were 24 required off the final over. Rinku Singh clubbed three boundaries and a six but came up one boundary short.
Lucknow and Punjab now join three other teams on six points, headed by Delhi Capitals, the only unbeaten team left.
