Leeds were still without top-scorer Jermaine Beckford so Malcolm Christie made his first league start for the club and Lee Trundle, who signed on loan from Bristol City on Friday, left the bench in the second half, along with Bradley Johnson on his return from a loan spell with Brighton.
Paul Thirlwell and Peter Murphy were passed fit for Carlisle, who still lacked Chris Lumsdon due to a lack of match fitness.
Bridges should have fired Carlisle ahead on seven minutes but, when put through by Cleveland Taylor, the striker saw his shot superbly saved by Casper Ankergren who was quick to leave his line..
Leeds' Argentine striker Luciano Becchio rustled the side-netting after a surging run by Robert Snodgrass, but the home side had to work hard for openings against opposition in the lower reaches of League One and managed by former academy coach Greg Abbott.
And it was Graham, who played three games for Leeds in 2006, who opened the scoring in the 27th minute.
Graham Kavanagh's excellent pass found Taylor on the right and he beat Alan Sheehan before crossing for Graham to stab home his 14th goal of the season.
Becchio wasted a clear heading chance and in the 35th minute Carlisle were two up when a mix-up in the Leeds defence allowed Graham to tee up a chance for Bridges, who made no mistake with a low shot from a narrow angle.
Leeds should have had a penalty when Fabian Delph was tripped by Jeff Smith but referee Kevin Friend waved play on.
Trundle, Johnson and Andy Robinson replaced Christie, Sheehan and Andrew Hughes for Leeds, while Carlisle sent on Scott Dobie for Bridges and Marc Bridge-Wilkinson for Taylor.
Becchio hit a post and Robinson missed an open goal in the immediate aftermath, but Carlisle should have grabbed a third when Bridge-Wilkinson fired wide from six yards.