Former England striker Peter Crouch added a last-gasp second as Mark Hughes' side saw off League One Rovers 2-0 at Prenton Park.
Irish midfielder Ireland, hoping to rejuvenate his career after moving on a season-long loan from Aston Villa, struck midway through the first half with Crouch striking with the last kick of the game.
Stoke should have won more comfortably and spurned numerous other chances, particularly in the first half.
They were almost made to pay as Jean-Louis Akpa Akpro and Chris Atkinson went close for Rovers in the second half but the Merseysiders were to rue the misses as Crouch had the final say.
Stoke had made seven changes, which included giving Ireland a run from the kick-off after two substitute appearances and a first start of the season for Crouch.
Hughes also rested first-choice goalkeeper Asmir Begovic to give Thomas Sorensen 90 minutes while Jermaine Pennant and Wilson Palacios were other inclusions.
Tranmere, having won just once and lying 19th in League One, were not expected to provide much opposition and even rested their veteran 40-year-old defender Ian Goodison with other matters pressing.
The Premier League side's superiority, despite the changes, was evident from the outset and, as appears to be becoming customary under Hughes, they passed the ball around effortlessly.
A first chance opened up when Austrian striker Marko Arnautovic, looking to confirm positive first impressions at the club, fired wide from 35 yards.
A succession of opportunities came along with Glenn Whelan missing the target, Ireland having a shot deflected wide and Crouch heading at Owain Fon Williams.
The opening goal came after 23 minutes as Arnautovic slipped in Ireland and the former Manchester City star stroked a fine low finish across Fon Williams into the far corner.
Tranmere offered little aside from a tame shot at Sorensen from Akpa Akpro after what had seemed a threatening run. Crouch almost let in Ryan Lowe after a careless pass but the Stoke defence recovered.
Geoff Cameron had Stoke's first chance of the second half when he shot across goal after good hold-up play by Crouch, who headed over soon after from a Pennant cross.
At times it seemed Stoke's play was too elaborate and more of their traditional direct style might have put the game beyond Tranmere earlier.
The home side produced one of their best moments after 56 minutes when Akpa Akpro cut inside from the left on a good counter-attack but curled his shot over.
Stoke were again almost punished when Lowe pulled back a deep Akpa Akpro cross but Atkinson sliced wide from a clear-cut opportunity.
Tranmere were growing in confidence and wanted a penalty when Lowe went down after a clash with Marc Muniesa but referee David Coote was not interested.
Substitute Jonathan Walters had two chances to finish the contest inside the last eight minutes but headed over and then shot wide after shaping to go round Fon Williams.
But Walters did make a telling contribution as time ran out, playing in Crouch and the forward delicately chipped over Fon Williams.