The big defender rescued keeper Matty Glennon as early as the sixth minute when the former Carlisle favourite could only parry a dipping Joe Garner drive. And Garner played a part in Carlisle's opener after 21 minutes by flicking on Marc Bridge-Wilkinson's near post corner for Livesey to lash the ball in at the other post.
It was the first goal Town had conceded in 486 minutes of football and Keiren Westwood had to dive full-length two minutes later to hold an Andy Booth header as they threatened a quick reply.
As an entertaining half drew to a close, Glennon palmed away a fizzing Peter Murphy free-kick, Michael Collins shot tamely at Westwood and the Carlisle keeper raced to the edge of his box to block Collins' progress when the Huddersfield man pounced on a mis-hit back pass by Evan Horwood.
The Terriers took the game to Carlisle from the restart and Westwood clawed a wicked cross by Robbie Williams from under his bar. Then Simon Hackney denied Huddersfield an equaliser with a goalline clearance from a Frank Sinclair header.
Garner fluffed a good chance to double Carlisle's lead in the 63rd minute, shooting across the face of goal when Graham's ball found him unmarked with just a few yards out and a combination of Page and Sinclair hustled Graham off the ball as he raced over the halfway line after a long clearance.
But the ex-Middlesbrough man grabbed his 11th goal of the season in the 76th minute. Glennon needlessly helped on a Garner header that looked to be going wide and, when Huddersfield failed to clear the resulting corner, Graham stabbed the ball home at the near post.