Barca's win will help partially erase the memory of last season's shock 2-1 defeat at Celtic Park but the result was in the balance before Brown's red card, despite the wealth of possession the visitors enjoyed as usual.
At first glance, the decision seemed harsh, with Celtic having men behind the ball when Neymar was hauled down in full flight. But replays showed Brown then flicked his foot out at the Brazilian, and the midfielder didn't appear to complain too strongly when French referee Stephane Lannoy brandished the card as Neymar rolled around on the ground.
Lennon was furious on the touchline at the time of the incident and again afterwards.
"It's a physical game and I don't think Neymar does himself any favors the way he behaves sometimes," said Lennon.
In earlier television interviews, Lennon had said that "Neymar has a reputation in the game."
Barcelona coach Gerardo Martino said he didn't see the incident in question, saying through a translator: "I don't want to give an opinion on something I couldn't see well."
From being a tight contest, Barca could finally escape the shackles with a man advantage and Sanchez delivered the most incisive piece of football of the whole match and Fabregas, playing as the central striker in Messi's absence, did the rest.
"This match has been the most demanding we have had to play so far (this season)," said Martino, whose team have started the season with a club-record seven straight wins in La Liga and the crushing victory over Ajax.
Barca stars Xavi Hernandez and Gerard Pique are among the world's star players to have talked about the spine-tingling atmosphere at Celtic on big European nights. "No words to describe the atmosphere at Celtic Park," Pique tweeted as he left the stadium last year.
And the visitors couldn't have failed to be impressed by the crowd's rousing rendition of "You'll Never Walk Alone" when both sets of players were in tunnel, or the huge roar as Celtic emerged from an on-field team huddle for the kickoff.
Every kick, tackle and attempted break by Celtic was received with an enormous din although it took the home side 1 minute 44 seconds to touch the ball for the first time as Barcelona slipped quickly into its passing groove.
Adopting the blueprint of most of the smaller teams playing Barca nowadays, Celtic had every man behind ball bar its lone striker, Georgios Samaras, when defending. It meant the visitors had all the possession 72 percent in the first half but few clear sights at Fraser Forster's goal.
The dynamic presence of Messi out with a right thigh injury was sorely missed, with Barca sometimes even ditching its tiki-taka passing and resorting to sending diagonal cross-field passes in behind Celtic's full backs to launch attacks.
Indeed, Neymar's clipped half-volley that flashed past the post from Dani Alves' right-wing cross was the only chance of note in the first half. Barca actually looked more dangerous and sprightly when Celtic was on the attack but that seldom happened.
The match became even more lopsided after Brown's silly sending-off, after which Neymar was vociferously booed whenever he touched the ball.
Still, though, Celtic defended comfortably and came close to taking the lead when first James Forrest brought a diving save out of goalkeeper Victor Valdes before Charlie Mulgrew headed the resulting corner narrowly wide.
But Barca was not to be denied and within minutes of coming on for Pedro Rodriguez, Sanchez crossed for Fabregas to head the ball back in the direction it came.
Forster had no chance but he produced heroics to deny Sanchez, twice, and Neymar as Barcelona broke at will late on.
