The Cumbrians were on the front foot from the first whistle and went ahead after a quarter of an hour when Graham despatched Scott Dobie's cross with a stunning volley into the roof of the net.
Four minutes later the ex-Middlesbrough striker made it two when he powered a header into the net from Simon Hackney's perfect cross.
But with Carlisle looking to run up a cricket score, Crewe edged back into the game with their first attack.
Julien Baudet cleverly headed back across goal allowing Chris McCready to nod the ball home from close range.
With that the home side slick passing game faded and Baudet was unlucky to equalise when his looping header struck the top of the bar with ex-Crewe keeper Ben Williams stranded.
After the break Graham should have netted his hat-trick goal but could only flash his shot wide through a crowded penalty area.
Joel Grant then somehow missed the chance to equalise when he poked wide from six yards before Carlisle made the visitors pay for slack finishing.
Graham's pass to Dobie was intercepted, but the Cumbria-born striker took advantage of the defensive slick and slammed a shot passed keeper Steve Collis from an acute angle.
Dobie then made way for substitute Michael Bridges and the ex-Premiership striker soon delivered a perfectly weighted pass for Hackney to slot past Collis.
With the Crewe defence crumbling, Baudet slipped, allowing Graham a free run on goal, but with just the keeper to beat he somehow screwed another hat-trick chance horribly wide.
But with five minutes to play Carlisle were left sweating when Williams flapped at a Grant cross and Tom Pope was on hand to take advantage and head into the empty net.