Alan Thompson's goal proved decisive for last season's UEFA Cup runners-up but the match, expected to be one of the European games of the season, was marred by a halftime tunnel incident that led to Celtic goalkeeper Robert Douglas and Barca midfielder Thiago Motta being dismissed.
Barca also had Javier Saviola red-carded while two of their Primera Liga rivals lost on their travels, Valencia going down in a 1-0 upset to Turkish side Genclerbirligi and Real Mallorca suffering a 4-1 drubbing by England's Newcastle United.
The four matches involving Spanish teams, including Villarreal's 2-0 home win over AS Roma, got the go-ahead from UEFA despite calls for their postponement following the Madrid bomb blasts that killed over 190 people earlier in the day.
They were preceded by a minute's silence with players wearing black armbands as a mark of respect for the victims.
Dignified scenes before kickoff at Celtic Park failed to be repeated at halftime when Celtic and Barcelona were reduced to 10 men.
Barcelona coach Frank Rijkaard said both sides should take blame for their aggressive behaviour.
"The (Barca) players felt the referee had made some strange decisions but that doesn't mean you have to over-react because that damages the team," Rijkaard told reporters.
"The players from Celtic were also little bit aggressive going towards the changing room and you can't only blame the players from Barcelona for this."
With tempers still running high, Saviola was shown a straight red three minutes after the break for kicking out at Thompson.
The Celtic midfielder then rubbed salt into the wounds by volleying in the game's only goal from Henrik Larsson's knockback to give the Scottish premier league leaders a narrow advantage to take to the Nou Camp.
Valencia, twice Champions League runners-up in recent years, and Mallorca boasted 100 per cent records away from home in this season's competition but could not extend them.
FIGHTING CHANCE
Ankara-based Genclerbirligi, who languish 10th in the Turkish first division, have knocked out Blackburn Rovers, Sporting Lisbon and Parma and gave themselves a fighting chance of reaching the last eight thanks to a 12th-minute penalty from Belgian Filip Daems.
Mallorca looked set to notch up another away victory when Fernando Correa netted the opener against Newcastle but the goal merely roused the hosts and strikes by Craig Bellamy, Alan Shearer, Laurent Robert and Titus Bramble should guarantee the English side a place in the last eight. The Spaniards also had Edu Moya dismissed late on.
Liverpool coach Gerard Houllier thinks his side will need no motivation for the return against Olympique Marseille at the Stade Velodrome after he was irritated by the French side singing in the dressing room after a 1-1 draw at Anfield.
"They think it's over because they are singing in the changing room," Houllier told reporters.
"I don't think it is over. The second leg will be played in a fortnight and we know we have to score and we know we can do that."
Striker Didier Drogba grabbed a precious away goal for Marseille from close range 12 minutes from time to cancel out a 55th-minute strike by Czech forward Milan Baros.
Three-times UEFA Cup winners Inter Milan ground out a goalless draw in Lisbon against Benfica while in other ties, Girondins Bordeaux of France overcame Belgium's Club Bruges 3-1 and Ligue 1 AJ Auxerre were held to a 1-1 home draw by Dutch side PSV Eindhoven.
The second legs take place on March 25.
