It was a massive victory for Swansea under new manager Roberto Martinez and boosted their hopes of a top-six finish.
Swansea always looked in control and held on well after going down to ten men in the 59th minute.
Top-scorer Trundle was dismissed for two crazy yellow cards. The first after 53 minutes was for over-celebrating his goal and the second on 59 minutes for taking a shot at goal after an offside decision had been given against him.
Robinson put City in front after only six minutes against his hometown club by heading in from a Leon Britton cross.
Trundle was unlucky to get a bad bounce of the ball after charging down a clearance kick and home keeper Gavin Ward was lucky to deflect a Robinson strike for a corner.
Tranmere gradually warmed to their task but Chris Shuker had one shot blocked and headed over with another effort.
There was a blow for Swansea when Kristian O'Leary was forced off after 36 minutes after an ankle injury.
Dennis Lawrence went on to play a big part in central defence with skipper Alan Tate moving into midfield where he was a big influence throughout.
Shuker saw a 25-yard free kick well held by Willie Gueret approaching half-time and Gareth Taylor headed wide.
Trundle almost got a second for Swansea four minutes after the break but saw his shot turned against an upright by keeper Ward.
But Trundle did strike in the 52nd minute. He raced clear from a through pass by Ian Craney and beat Ward with an angled effort.
Tranmere rallied in search of a point but both Chris McCready and Taylor headed wide.
Then John Mullin saw a well-struck shot pushed over by keeper Gueret in the 85th minute.
Chris Greenacre tried hard for Tranmere but they were strangely out of sorts as they fell to their third home defeat of the season.