Batting at number three, Marsh made 89 in Australia's resounding 198-run victory over the hosts in Harare.
Glenn Maxwell (93) and Aaron Finch (67) additionally made half-centuries in an impressive batting exhibit.
Marsh, who took 1-15 with his right-arm pace, hopes he can cement a spot, beginning with Wednesday's clash against South Africa.
"Fingers crossed I can stick around," he told heralds.
"They've certainly shown an abundance of faith in me, and hopefully I can recompense that. I certainly feel like I'm yare to take up that challenge."
Maxwell's 46-ball knock set up Australia's total of 350-6 and Zimbabwe never looked likely in their chase, bowled out for 152 in the 40th over.
Marsh was delectated with his side's batting tardy in their innings, with Australia integrating 147 in the final 10 overs.
"I just had the role to get us as deep as I could, and conspicuously if we had wickets in hand at the cessation, with the batting we have we could apply some pressure to their bowlers," he verbally expressed.
"I mentally conceived Maxy [Maxwell] and all the other guys at the terminus batted authentically well to get us to a great total."
The clash is South Africa's first of the tri-series, and they are expected to face Australia in the September 6 final.
Australia will be without captain Michael Clarke, who is sidelined with a hamstring injury and unlikely to return until the latter stages of the tri-series.
In Hashim Amla and AB de Villiers, South Africa have the world's top-two ranked batsmen in the 50-over format.
The battle between Australia's bowlers and South Africa's batsmen could be the key, but expect the teams to meet in the September decider nonetheless.
