The tourists warmed up for their first Test against England by finishing the first day of three on 348-9, thanks in large part to Marshall's well-taken century.
Marshall's 128 came on the same day Hamish, who after limited chances with New Zealand has concentrated on country cricket, brought up a three-figure score for Gloucestershire 166 miles away in Bristol.
With New Zealand set to face England in their first Test on May 15, it was James Marshall's display that has more significance.
"It's nice to get a hundred early in the tour. I was just pleased with the way I constructed the innings on a surface that wasn't easy to bat on," said James.
"There were a lot of shots that probably weren't on today and you had to be mentally strong not to bring those out."
Cook was hurt when he dropped Daniel Vettori at gully off Ryan ten Doeschate, but after an X-ray he is expected to return to duties on day two and against the Kiwis in the Test series.
Ten Doeschate took 5-57, underpinning Marshall's 128 as the cornerstone of a an innings which saw contributions notably from Aaron Redmond (35), Daniel Flynn (39) and Iain O'Brien.
Redmond hit seven boundaries in his breezy 35, but after Jamie How was lbw to Tony Palladino Ten Doeschate removed Redmond in similar fashion in his first over.
Marshall and Flynn then put on 128 in 25 overs, before the latter drove Ten Doeschate to mid-off, while Jacob Oram, Vettori and Kyle Mills did not last long.
Marshall ploughed on until he gave Ten Doeschate his fifth wicket.
New Zealand rallied with Michael Mason and O'Brien frustrating Essex by adding an unbroken 52 late on.