Marsh hit 114 and put on a 181-run partnership with Chris Rogers (84) ??? his rival for a spot in the first Ashes Test ??? for the first wicket.
Steve Smith (71no) and Michael Clarke (56) also passed fifty, as the tourists??? top order all got decent time in the middle.
Kent???s Matt Hunn was the pick of the bowlers, with the 21-year-old seamer taking 3-78.
Marsh replaced Rogers in Australia’s recent series win in the West Indies after he was concussed while batting in the nets and the two batsmen now seem to be vying for same openers spot in the Test team alongside David Warner.
They were sent out to bat together after Kent captain Sam Northeast won the toss and elected to bowl first.
Rogers ??? who made headlines earlier this week after a company he co-founded was blocked from selling hospitality packages for the sell-out Lord’s Ashes Test ??? was dropped in the slips on 21 by Adam Riley off Mitchell Claydon.
It was the only chance Rogers offered though and he seemed set for a century, before falling 16 runs short, lbw to Hunn.
But Marsh pressed on and did go through to his hundred, off 172 balls, but shortly after tea he holed out to deep-point off Hunn.
From 242-2, Smith and Clarke built another formidable partnership, adding 95 for the third wicket.
Smith ??? the number one ranked Test batsman in the world ??? brought up his half century from 80 balls, with four fours and a six.
But Hunn struck again shortly before the close as Clarke was caught down the legside by Sam Billings.
Away from the middle, a bizarre incident occurred in the final session as a spectator pitched a one-man tent on the outfield, emerging in a pair of shorts and flying helmet, before being escorted from the field by security.