Bridges came on as a late substitute and fired low under the body of diving goalkeeper Josh Wagenaar from 25 yards.
Carlisle had looked bright early on with Cleveland Taylor dangerous on the right but his crosses came to nothing and they began to run out of steam.
Yeovil survived that early onslaught and took the lead after 31 minutes when Danny Schofield and Gavin Tomlin carved an opening and Paul Warne flicked a pass to Gary Roberts for the final touch.
Both sides created chances in the second half and had appeals for penalties turned down for handling offences.
But Carlisle always looked dangerous and although there were question marks over the goalkeeper for the equaliser, Wagenaar had earlier made a superb diving save from a Graham Kavanagh free-kick.