The visitors had to leave two players behind at their hotel and manager Sean O'Driscoll claimed four others who took to the pitch were not 100 per cent.
The only goal came after just six minutes when Gareth Taylor headed on a free-kick by Paul McLaren and Greenacre stole in to head past Ben Smith from close range.
Tranmere went on to create the better chances and Chris Shuker shaved the outside of an upright with half-time approaching.
In the second period, McLaren lobbed over and Carl Tremarco brought an excellent finger-tip save from Smith in the 78th minute.
The Merseysiders made light of losing two players through injury. Loan signing John Thompson was forced off after only 25 minutes with a leg injury, but Shane Sherriff proved an able deputy and never put a foot wrong.
Kevin Ellison was the other injury victim after an hour when Danny Harrison took over.
Doncaster never gave up the fight but they failed to seriously test home keeper John Achterberg.
The keeper made easy saves early on from efforts by Graeme Lee and Adam Lockwood, while the latter saw another chance after a corner blocked out.
Doncaster continued to struggle against a solid Tranmere rearguard in the second period with their best efforts coming from Brian Stock who fired wide and Lee who also missed the target with a header following a corner.
The visitors sent on Lewis Guy in place of Paul Green for the closing stages to try to put more emphasis on attack, but it turned into a frustrating afternoon for them as they fell to a sixth away reverse.
Greenacre was always busy for the winners and McLaren and Jason McAteer were steady in midfield.
Stock was best on the day for the losers, but former Tranmere player Gareth Roberts and Lockwood were also prominent.