Marsh brought up his patient hundred off 213 balls in 328 minutes as Australia rumbled past 400 in their first innings after being sent into bat by England captain Joe Root.
Australia's score surpassed their previous best of 383 in the first innings of last year's pink ball Test against South Africa in Adelaide.
At the dinner break, the home side were 409 for seven with Marsh on 103 and Cummins, who took 37 balls to get off the mark, not out 44.
Marsh and Tim Paine both vindicated their contentious Test recalls as Australia took charge.
Marsh, in his eighth recall to the Australian team, backed up his 51 in Brisbane to strengthen Australia's position in the first-ever Ashes day-night Test.
Wicketkeeper Paine, who received a surprise Ashes recall after being out of the Australia side for seven years, cracked 57 off 102 balls.
Paine, who took a couple of painful blows on his right index finger which he shattered seven years ago, shared in an 85-run stand with Marsh before he was caught on the boundary pulling Craig Overton.
Paine had eight pins and a metal plate inserted into the fractured finger which caused him to lose his spot in the Test side back in 2010.
England had several calls go against them on day two. Marsh had an lbw decision on 29 by James Anderson overturned when a review showed the ball would have gone over the stumps.
Anderson was frustrated again, much to the delight of the home crowd, when a review denied him again after Paine on 24 had been given out lbw on the field.
Peter Handscomb only lasted three balls at the start of the day before he was trapped leg before wicket playing back on his overnight score of 36 by a fired-up Stuart Broad, who gave him an animated send-off.
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