Celtic had never won in Spain in 12 previous attempts but gave Barca, chasing a third European crown in five years, a mighty scare when they took an 18th-minute lead at the Nou Camp through Giorgos Samaras.
The home side dominated possession but were struggling to penetrate Celtic's massed ranks until Andres Iniesta combined with Xavi and squeezed a shot just inside Fraser Forster's post moments before half-time.
Barca pressed as Celtic tired in the second half and after Forster twice denied a typically hyperactive Lionel Messi and David Villa struck the frame of the goal, full-back Alba popped up at the far post in the 94th minute to deflect Adriano's cross over the line.
"It was a move involving the whole team and I just had the good fortune to be in the right place," Alba said in an interview with Spanish television.
"I don't know what they [the Celtic players] eat but they are very powerful in the air, in defence as well as attack," the Spain international added.
"They really closed ranks at the back and didn't give us any space but we still managed to create a lot of chances."
It was a bitter blow for Celtic and their vocal travelling support high up in the stands, although the Scottish champions remain second in the group on four points, with Barca on nine and Spartak Moscow, who won 2-1 at home to Benfica, on three.
Neil Lennon's side were chasing the ball for long periods, Barca had 76 percent possession, and while they at least showed some threat on the break they only managed one shot on target to Barca's 10.
"We conceded ground when we had to, but we defended brilliantly and at times, on the counterattack, we looked dangerous," Lennon told a news conference.
"While I'm disappointed for the players I'm also immensely proud of them. They were brave and stuck to what we asked them to do," he added.
"We're playing it game by game and although we didn't budget for anything from tonight we came very, very close."
MESMERISING EXCHANGE
With Carles Puyol and Gerard Pique injured, Barca coach Tito Vilanova deployed 21-year-old Marc Bartra at centre-back, only his third appearance in the competition, while Alex Song replaced the suspended Sergio Busquets in midfield.
Celtic were immediately on the back foot and Iniesta sent Alexis Sanchez clear in the second minute but the Chilean skewed his shot narrowly wide with only Forster to beat.
The Englishman then produced a fine save to keep out Bartra's header but it was the visitors who grabbed a surprise lead through Samaras, who leaped to head an inswinging free-kick towards goal and the ball deflected off Javier Mascherano and spun past Victor Valdes into the net.
Samaras went off injured shortly before the break after twisting his ankle.
Iniesta's equaliser was the result of a mesmerising exchange of passes between the Spain playmaker, Messi and Xavi, with Iniesta somehow finding the space in a crowd of defenders to get his shot away.
