The 27-year-old, on loan from Aston Villa, netted his first goal in almost two years after being handed a first start for the Potters, with Peter Crouch doubling the advantage
The midfielder had not found the back of the net for 634 days since scoring for Aston Villa at Chelsea, but he struck in the 22nd minute at Prenton Park to help book Stoke’s place in the fourth round.
Tranmere threatened in the second half as Stoke began to tire, but their finishing failed to test visiting goalkeeper Thomas Sorensen, before Peter Crouch added a last-minute winner to put a gloss on the result.
Mark Hughes made seven changes to the side beaten at Arsenal on Sunday with Ireland handed his full debut and Crouch chosen ahead of Kenwyne Jones in attack. The hosts rested experienced defender Ian Goodison and Joe Thompson, with Ash Taylor and Cole Stockton coming into the side.
The visitors started well and posed more of a threat than their League One opponents, although Owain Fon Williams was not tested until the 20th minute.
Arnautovic and Glenn Whelan sent long range efforts wide of goal, but Crouch did call the keeper into action with a close-range header that Williams saved easily.
However, Stoke soon snatched the lead as Ireland broke the deadlock with his first strike for the club; the on-loan midfielder received a perfectly weighted pass from Marko Arnautovic inside the penalty area and fired across Williams into the far corner of the Welshman’s goal.
Hughes’ side continued to pressure their hosts and the Austrian threatened again with a low shot that was just wide.
Jean-Louis Akpa Akpro, meanwhile, had Tranmere’s best opening of the half but his effort was deflected wide of Sorensen’s goal.
The Ivorian striker had the first chance after the break too but his finishing let him down after excellent approach work. He picked up the ball inside the Stoke half, beat a couple of defenders but could only fire over from inside the penalty area.
Ireland lasted an hour before he was replaced by Oussama Assaidi for his second Stoke appearance after joining on-loan from Liverpool.
The hosts continued to increase the tempo as Hughes’ side began to tire, and in the 68th minute Chris Atkinson should have done better with a good chance; Ryan Lowe and Akpa Akpro combined well on the edge of the penalty area but Atkinson shot across goal and wide without troubling Sorensen.
Two visiting substitutes nearly combined to secure the victory with seven minutes to play, but Jonathan Walters wasted his first sight of goal.
Assaidi teased the Tranmere defence, beat his full-back on the left-hand side and delivered a perfect cross to the back post but the Ireland international headed over the crossbar.
The Moroccan was causing problems to the hosts and he created Stoke's second goal with the last kick of the game, his cross converted by Crouch to secure their win.