Striker Leigh Griffiths lit up a dull first half at the Constant Vanden Stock Stadium with a poacher’s goal in the 38th minute and five minutes after the break, Patrick Roberts’ drive took a deflection in off substitute Kara Mbodji.
Anderlecht’s interim boss Nicolas Frutos stood on the sidelines frustrated as Brendan Rodgers’ side withstood everything the Belgian champions could throw at them before Scott Sinclair scored a breakaway third in added time.
Celtic lost 5-0 at home to Paris St Germain on match day one with Anderlecht going down 3-0 to Bayern Munich in the other Group B game.
If nothing else, the Glasgow club’s second-ever away win in the competition — their last was against Spartak Moscow in October 2012 — takes the Scottish champions a long way to ensuring European football after Christmas.
It was was a hugely significant result for Celtic as they had failed to register a win in the group stages last season, the first under Rodgers.
Afterwards, the manager said: “It was an outstanding performance and result. We gave the ball away loosely at times and needed to be cleaner in our build-up play.
“There are still areas to improve but they were exceptional. I think you can see the fitness of the players. For a Scottish team to come away and get a 3-0 victory and keep a clean sheet at this level is a big achievement.”
Around 1,000 Hoops fans were inside the 21,000-plus capacity stadium with many others having travelled in hope of getting a ticket and they helped provide a raucous atmosphere.
There was nothing particularly thoughtful or composed about the early stages and indeed the well-worked opener was at odds with what had gone on before.
Midfielder Oliver Ntcham, struggling to get his game going, released Kieran Tierney down the left and when the full-back drilled the ball across the six-yard box Griffiths gleefully knocked the ball into the empty net for his eighth of the season.
Seconds later, with the celebrations of the Celtic fans still ongoing, Parkhead goalkeeper Craig Gordon had to make a decent save from captain Sofiane Hanni’s 25-yard drive.
Just before the break, the Belgian outfit suffered another blow as Mbodji replaced Uros Spajic after the defender failed to shrug off an ankle knock picked up earlier.
It was the Anderlecht skipper who had Gordon in trouble with a powerful drive at the start of the second half, the Scotland keeper unable to hold it before Lukasz Teodorczyk was flagged offside as he waited on the loose ball.
However, the home side paid for some slack play by Olivier Deschacht which saw Roberts take possession, and he turned back inside and when he fired in a drive from 16 yards it came off Mbodji and past Boeckx.
That seemed to sap confidence from Anderlecht although Hanni had another effort, driving just over the bar.
At the other end Roberts came close to making it three with a shot which just evaded the goal and substitute Callum McGregor, on for Tom Rogic, dithered in the box when he should have got his shot away.
As Anderlecht threw everything forward in an attempt to get back into the game, Gordon stretched in the 74th minute to tip a goal-bound drive from Pieter Gerkens over the bar. Thereafter Celtic managed the game with some assurance, having the hunger to break with pace in added time, with Ntcham setting up Sinclair to steer in the third.
Boeckx, Chipciu, Spajic (Mbodji 44), Dendoncker, Deschacht (Stanciu 58), Appiah, Hanni (Bruno 70), Gerkens, Trebel, Onyekuru, Teodorczyk.
Gordon, Lustig, Simunovic, Boyata, Tierney, Roberts (Forrest 77), Ntcham, Brown (Bitton 71), Rogic (McGregor 65), Sinclair, Griffiths.
Jesus Gil Manzano (Spain).