Chris Greenacre robbed Carlisle captain Paul Thirlwell and fired a great 50-yard ball across the pitch to Chris Shuker on the left wing. United's Paul Arnison should have cleared but lost the flight of the ball, allowing Shuker to lay an inch-perfect cross for Curran to volley into the roof of the net from close range.
Danny Graham should have given the Cumbrians the lead after only two minutes but, despite rounding goalkeeper Danny Coyne, he delayed his shot allowing Ian Goodison to make a good block on the line.
Tranmere lost their burly striker Calvin Zola after just 17 minutes when he chased on to a through ball and pulled up with what appeared to be a hamstring injury. But the loss of Zola - without a goal this season - was balanced by the introduction of the lively Curran.
Carlisle playmaker Marc Bridge-Wilkinson was the only bright spot for the home side and he almost found the net with a beautifully lofted curling shot from the edge of the box which narrowly missed Coyne's post. Steve Jennings should have done better when well placed but scuffed a poor shot at Keiren Westwood and the Carlisle keeper proved his worth again with a great double save.
Curran found room to turn 20 yards out and when his shot was well saved, Robbie Stockdale was on hand but again Westwood saved the day with a crucial block. Substitute Simon Hackney appeared to be flattened in the box by Antony Kay but the referee waved play on and Danny Carlton almost took advantage of a defensive error but Coyne was out in a flash to give Rovers all three points.