The Cumbrians maintained their formidable home form with an amazing 14th successive home win to extend their club record.
Livesey's winner, four minutes after the break, came from a perfectly worked corner from the left foot of Simon Hackney.
Peter Murphy created problems with his clever near post flick on and Livesey took full advantage to smash into the roof of the net from point-blank range.
Five minutes from time a shocking error from the Cobblers on-loan defender Mark Little sealed the points for Carlisle.
Little's poor back-pass to keeper Mark Bunn never looked likely to reach him on the edge of the box, allowing Danny Graham to intercept.
He immediately cut the ball back to Marc Bridge-Wilkinson and the ex-Port Vale midfielder calmly slotted past the covering Liam Dolman on the line.
The home side should have been out of sight well before then and their slack finishing almost allowed Town to grab an unlikely equaliser on 80 minutes.
Substitute Jonathan Hayes broke away and slotted a neat pass to Giles Coke, but with a the goal seeming certain, Keiren Westwood leapt off his line to block with his legs.
A goalless first half saw Graham go closest to breaking the deadlock as early as the third minute, when his shot from a tight angle hit the outside of Bunn's post and flashed across the goalmouth.
Chris Lumsdon and Hackney both went close for Carlisle, before Coke finished poorly for the visitors after a thrilling run through the heart of the home defence.
Graham should have made it 2-0 on the hour, but he somehow flicked wide from Hackney's exquisite lofted pass.
And Hackney fluffed his lines when he shot tamely at Bunn from close range, but the result was never really in doubt as Carlisle pressed towards a third promotion in just four seasons.