The Cumbrians made little headway against the play-off chasing visitors and can count themselves extremely lucky to have only conceded twice in a match that did little to convince the fans that their side will pull away from the threat of relegation.
David Perkins wrapped up the points for the U's late on and the win was never seriously in doubt after Livesey's bizarre strike.
The towering centre-back inadvertently started and finished the move that opened the scoring for the visitors in the 11th minute.
First he gave away a sloppy square pass to the dangerous Mark Yeates, who found Lewis Gobern and the winger's cross was accidentally turned in at the back post by Livesey as he attempted to clear.
Ten minutes later, Livesey was given a lucky reprieve when he pulled down Gobern in the box for what looked like a certain penalty but referee Clive Oliver thought otherwise and waved play on .
Six minutes before the break the hosts almost snatched an undeserved equaliser but their former on-loan full back Marc Tierney was on hand to clear acrobatically off the line.
Cleveland Taylor mis-hit a cross that was tipped onto the bar by Jimmy Walker and with the ball bouncing across the line and Danny Graham set to nod in, Tierney somehow got there first to clear.
From the resulting corner, Graham leapt highest to bullet a header that was palmed over by Walker - the last action the visiting keeper was forced into.
With Evan Horwood on as a half-time substitute for the hapless Michael Liddle, Carlisle managed to turn in a slightly improved second-half performance, but with eight minutes to play the game was out of sight for them.
A neat one-two between Yeates and Scott Vernon found the latter clean through on goal but when the shot was blocked by Ben Williams, Perkins was on hand to slide the ball in from ten yards as three Carlisle defenders stood ball watching.