The hosts looked to be in total control at half-time thanks to goals from centre-back pairing Danny Livesey and Murphy.
But a spectacular strike from David Bell on 51minutes gave the Hatters all the encouragement they needed to push forward for what would have been a deserved equaliser.
On a pitch left little more than a mudbath and a near gale-force wind to contend with, it was Carlisle who dealt better with the conditions in the first half.
Their supremacy was rewarded on 22 minutes when captain Livesey volleyed in Marc Bridge-Wilkinson's corner for his second goal in four days.
Seven minutes later, Evan Horwood almost netted when his cross was caught in the wind and Luton keeper Dean Brill did well to tip over the accidental effort.
Seconds later, Brill made two great saves to deny Grant Smith and Simon Hackney before blundering moments later to gift Carlisle a second.
A looping cross from Chris Lumsdon held up in the wind, but Brill was determined to get to the ball nearly 15 yards off his line.
But his effort was in vain as Murphy arrived first to head into the open net.
A cricket score looked likely for the Cumbrians, but a more solid Luton turned out for the second half and Bell provided their lifeline.
Side-stepping a defender on the edge of the box, the Luton dangerman pulled the trigger with a low left-foot drive from 30 yards to beat Keiren Westwood at his near post.
With Carlise struggling to deal with the wind, the visitors piled on the pressure with Paul McVeigh, Keith Keane and Bell again all going close.
But the hosts held on for their 13th successive home win and left Luton closer still to the relegation trap door.