Tranmere lived dangerously late on but over the 90 minutes just deserved the verdict.
Ian Goodison made a big impact in his first outing after a two-match suspension and there was also a lively contribution from Edrissa Sonko.
Joel Porter saw an early United effort blocked and, at the other end, a snapshot by Sonko was well held by visiting keeper Adam Lee-Barrett.
The only goal came in the 23rd minute. Sonko flicked on a long clearance by Ben Chorley for Curran to find the target in off the left-hand post with a fine strike from the edge of the box.
The visitors responded well and it needed a fine tackle by Andy Taylor to block out James Brown.
Then a Gary Liddle effort was somehow hacked clear by Chorley.
The visitors were unlucky not to level three minutes before the break when home keeper Danny Coyne denied Liddle with an outstretched leg.
Tranmere should have doubled their lead straight from the restart but Sonko mis-hit his shot when set up by Godwin Antwi.
Curran was denied a second goal through a marginal offside decision but just ahead of the hour.
Goodison headed into the side-netting on 62 minutes and soon afterwards Sonko brought an excellent response from Lee-Barrett after a good run.
Tranmere were looking nervous in the closing stages but some desperate tackling and some wayward shooting by United saw them hold on to their slender advantage.