The Bees quickly lost a first-half lead after Nathan Redmond cancelled out an opener from Jota but continued to look the better side and grabbed all three points when Alex Pritchard converted a 71st-minute penalty.
The visitors wasted a wonderful opportunity to take the lead in the first minute when Andre Gray found himself in the clear as the hosts waited in vain for an offside flag.
The former Luton man had all the time in the world but the ball appeared to take a bobble as he prepared to shoot and his effort went horribly high and wide.
A lively start to the game then saw Gary Hooper foiled by Brentford goalkeeper David Button at the near post before a snap shot from Stuart Dallas at the other end went just over.
The Bees were the better side in the early stages and it was no surprise when they took the lead after 21 minutes. Steven Whittaker carelessly gave the ball away on the edge of the visitors’ area and a swift break ended with Gray slipping the ball through for Jota who fired across John Ruddy and into the back of the net.
Norwich responded quickly and were back on level terms within six minutes. Redmond collected a deep cross on the right and cut inside before sending in a low shot that took a slight deflection off James Tarkowski and deceived the unlucky Button.
Brentford suffered another blow on the stroke of half-time when the influential Toumani Diagouraga was caught by a high challenge from Bradley Johnson and required prolonged treatment on the pitch.
The midfielder was fit to resume after the break, albeit with heavy strapping on his right thigh.
After a scrappy start to the second half Brentford finally provided a brief moment of excitement when Tarkowski had another chance to put Gray in but his pass was woefully misplaced.
Dallas then strung Ruddy’s palms from 25 yards as the home side struggled to create anything at the other end.
With the game entering its final quarter, it took a superb one-handed save from Ruddy to deny Jonathan Douglas as Brentford continued to look the better side.
They finally got their reward after 71 minutes when a combination of Carlos Cuellar and Michael Turner brought down Gray just inside the box and Pritchard made no mistake from the spot.
The Canaries were stunned but almost equalised with their next attack when Johnson’s audacious long range effort came back of the crossbar before Button produced a superb reflex stop to prevent Cameron Jerome from tucking away the rebound.
The keeper made another fine stop to deny substitute Lewis Grabban in injury-time as the Bees held out for a well deserved win.