The Bees goalkeeper denied Joe Garner from the spot after centre-back Tony Craig had been penalised for a hotly-contested aerial clash with the visitors' David Symington.
The Londoners broke the deadlock in the 20th minute, Jonathan Douglas hammering home a far post volley from Harry Forrester's deep corner, but the Cumbrians were level minutes later when Garner cleverly made space for himself after Harlee Dean had slipped to fire past Moore.
Moore kept his side in it just before the break when he parried Bradley Potts' 30-yard screamer into the path of Jon-Paul McGovern, whose first-time effort lacked venom.
Clayton Donaldson curled an effort just wide for Brentford after the break and then held his head as he powered a header wide.
Adam Forshaw twice went close for the hosts before Paul Hayes grabbed the winner, reacting quickest in the box to mis-hit the winner into the far corner, while the visitors also survived a late penalty claim, when Dean's header appeared to have been palmed to safety by a player in a melee.