The rugged defender, who came on at half-time for Peter Murphy, poked home from close range after Zigor Aranalde's free-kick was blocked for the only goal of the game.
The Cumbrians hadn't won at Millmoor since 1983, losing seven of their last nine fixtures against Rotherham and almost went behind after just three minutes.
Carlisle, seeking a seventh away win this term, came out of the blocks slowly but went close on 13 minutes with Kevin Gall drilling an angled volley wide following David Raven's free-kick.
A low-key spell followed although Danny Livesey had to be alert to clear a dangerous cut-back from Chris O'Grady with loan stars Danny Graham, who had struck six goals in his previous five games, and Joe Garner kept under lock and key by the Millers' back four.
Westwood then had to be at his best to turn away Woods' sweetly struck low shot at full stretch as the hosts pressed for an opener.
With play-off rivals Oldham surprisingly losing to Cheltenham, Carlisle had all the impetus they needed to push on backed by a big travelling support, and Graham almost fired them ahead with a low shot that was blocked by Cutler after an error from Pablo Mills.
The Millers then went desperately close with Stephen Brogan's cross evading Henderson and arriving at the feet of Delroy Facey, whose effort was diverted in the nick of time by Jeff Smith with the goal gaping.
Fierce appeals for handball against Zigor Aranalde were turned down by Cumbria-based referee Mike Pike while Woods fired a rasping shot straight into the midriff of Westwood.
At the other end impressive keeper Westwood raced out well to smother Marc Newsham's effort and the on-song keeper then denied O'Grady.
Despite late pressure from the hosts, who laid siege to the Carlisle goal in the final ten minutes, the Cumbrians held out to net a priceless win.