The Lisbon club are level on seven points with the Scots but have a better head-to-head record. If Benfica's result at already-qualified Barcelona at least matches Celtic's at home to Spartak Moscow they will reach the last 16.
Their 20-year-old Liberian-born winger John, signed from Dutch side Twente Enschede in the close season, drilled home the opener after seven minutes but although Celtic surrendered most of the possession they were deadly up front.
Striker Giorgos Samaras equalised with the visitors' first opportunity, towering above the defence to head in from a corner after 32 minutes.
Benfica missed several chances as Celtic tried to hang on for the draw that would have put them through but their persistence paid off when Argentine Garay volleyed the winner after 71 minutes.
"It was not easy but it was deserved. We need to follow this path against Barcelona and I feel we can do a huge game there and get a result," said match-winner Garay.
Barcelona had qualified earlier on Tuesday with a 3-0 win at Spartak, who are bottom.
"I'm disappointed with the two goals we conceded," Celtic manager Neil Lennon told the BBC. "There was a lack of composure, a nervousness about us at times on the ball."
John chose the perfect moment to grab his first goal for Benfica.
Celtic squandered possession and Argentine winger Eduardo Salvio surged past two defenders on the right before sending over a cross that eventually fell for John to fire in.
"LUCKY" CELTIC
Benfica coach Jorge Jesus opted for the tall trio of Nemanja Matic, Luisao and Oscar Cardozo to counteract Celtic's danger from set pieces but the defence was still found wanting.
The Scottish champions equalised from their first corner just past the half-hour mark when Samaras was left unmarked at the far post and dispatched an easy header to the despair of the home fans on a wet evening in Lisbon.
"The game was really close, we lost it on details but the team effort was there and we'll look forward to the next game," Celtic striker Samaras was quoted as saying on the UEFA website. "We must stay positive, we can still reach the next round and we must focus on beating Spartak Moscow."
Benfica created several chances in the second half, the best when Brazilian striker Lima ghosted past a defender and then beat Celtic keeper Fraser Forster only for his shot to be cleared off the line by Welsh defender Adam Mathews.
"We beat a team who have been lucky in the Champions League," said Benfica coach Jesus. "They scored from a set piece and should had left here with three or four goals in their net. We will go into the match at Barcelona with a lot of confidence."
Celtic, who earned a memorable 2-1 win at home to Barcelona two weeks ago, ran out of luck with 20 minutes remaining. Matic floated a beautiful cross into the box and Luisao cushioned a header for Garay to volley into the net.
