Portuguese striker Rui Pedro ended the qualifying hopes of his countrymen during a scintillating opening first half in which the beleaguered visitors also had Douglao sent off.
Rui Pedro had his treble within 33 minutes and, while Alan had pulled a goal back on 17 minutes, Cluj were in full control by the break once Douglao saw red in added time.
While Cluj failed to add to their tally, victory ensured they moved level with Galatasaray, who beat already-qualified Manchester United 1-0 in Istanbul, with a game remaining in Group H.
The Romanians face United at Old Trafford on the final matchday with Gala travelling to Braga.
There was no intention to sit back in the opening exchanges in Romania as Cluj raced into a 2-0 lead after just 15 minutes thanks to Rui Pedro.
He was indebted to a sublime flick from Rafael Bastos for his first as the deftness in his pass cut open the away defence to allow Rui Pedro to race onto the ball and slot home.
The hosts were 2-0 up moments later when Modou Sougou's rampaging run down the right forced a hasty clearance which fell straight to Rui Pedro inside the box to lash home.
Braga threatened to get back into the game when Alan converted following Ruben Micael's searching cross-field ball with 17 minutes on the clock.
But that was the extent of their rally as Rui Pedro took over again as he completed his hat-trick just after the half-hour after turning in Sougou's pass from the right wing.
Braga's task then became infinity harder when Douglao saw red in injury time for violent conduct which left his team facing the prospect of a second 45 minutes with a man less.
The visitors managed to prevent any further damage but their appetite for the contest had long passed as Cluj cruised to their victory.
Beto made a brilliant stop to tip away Bastos' shot on 58 minutes while Mario Camoro also hit a post with a shot from 18 yards but it mattered not as both sides welcomed the final whistle.