The game was decided in the Dons’ favour by an 81st-minute goal from substitute Jabo Ibehre, who turned in a Jemal Johnson cross from close range.
Chris Shuker, Gavin Gunning and Craig Curran had good early efforts for the hosts but the Dons, content to soak up pressure, were not without their chances as David McCracken and Jason Puncheon missed the target before Luke Chadwick blazed over.
Tranmere went close after 51 minutes but Ian Thomas-Moore’s 20-yard strike rebounded off a post, with the frontman out of luck again 60 seconds later as he then struck the underside of the bar.
At the other end, Luke Daniels did well to hold onto a powerful drive by Puncheon but was lucky when Jermaine Easter fired over from 10 yards, before himself hitting the bar just short of the hour mark.
Ibehre finally grabbed the breakthrough with the game in its closing stages but Tranmere almost claimed a leveller in added time when Sol Davis headed off the line from substitute Terry Gornell.