Chris Martin celebrated the signing of a new three-and-a-half-year contract by firing home a 77th-minute winner for the Canaries, who moved above long-time leaders Leeds on goal difference as a result.
The visitors gradually got a foothold in the game and were handed a massive boost five minutes before the break when Holt saw red for an over-the-top challenge on midfielder Toumani Diagouraga.
To their credit the Canaries came out looking for a winner in the second half, despite being down to 10 men, and they finally found it when substitute Oli Johnson jinked into the box down the left before playing in a clever pass for Martin to slide the ball under the keeper.
Brentford, who had earlier hit the woodwork through Sam Wood, pressed for an equaliser and went desperately close after 82 minutes when Fraser Forster was forced into a superb save to keep out a low shot from Kevin O'Connor, before Myles Weston carelessly slid the rebound wide.
Norwich held on to make it 10 straight home League wins, equalling their all-time record for a single season.