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The Merseysiders made much of the early running but John Welsh had no luck with three efforts, while at the other end Liam Sercombe blasted well over from range.
Tranmere should have gone in front two minutes before the break but Ryan Brunt fired over in front of a gaping goal after being set up by the busy Welsh.
But it was 1-0 four minutes after the interval with sub Joss Labadie beating Lenny Pidgeley with a fine angled strike from 20 yards. The same player was unlucky to hit the bar with a 30-yard free-kick three minutes later.
The woodwork came to Tranmere's rescue after 71 minutes with James Dunne striking a post for City with a fine effort.
But Rovers made sure in added time with a breakaway goal by sub Jake Cassidy.
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