The visitors arrived needing to win by three goals or more to ensure it'd be they, and not Celtic, playing European football after Christmas but their efforts proved in vain.
An own goal from Jozo Simunovic, one of five changes to the Celtic side that thrashed Motherwell at the weekend, was all the Belgians could muster despite long spells of dominance, particularly in the first half.
And the Hoops held on to ensure their European adventure continued by finishing behind Paris-Saint Germain and Bayern Munich in the pool.
Anderlecht started quickly and had a glorious chance to open the scoring with less than three minutes on the clock.
Henry Onyekuru managed to pick out an unmarked Sven Kums in the penalty area but his side-footed shot was parried to safety by goalkeeper Craig Gordon.
Anderlecht proceeded to dominate, outplaying a tentative Celtic side, with Onyekuru out on the left causing Mikael Lustig all sorts of problems.
Adrien Trebel tested Gordon with an acrobatic volley after being played in Pieter Gerkens and then tried his luck with a fierce shot from outside the box that the Celtic goalkeeper was equal to.
Celtic, in stark contrast, offered little in the way of an attacking force, the returning Moussa Dembele struggling to hold the ball up and the Scottish champions will have been relieved to make it half-time with the game still goalless.
The hosts made two changes at the break, bringing on Tom Rogic and Olivier Ntcham and looked far stronger with the Frenchman firing wide within minutes of the restart.
Celtic continued to improve, Dembele making a fine run into the box but only managing a tame shot at goal before goalkeeper Frank Boeckx tipped a James Forrest shot around the post.
Despite being on top, Celtic conceded after 62 minutes, a hopeful cross by Dennis Appiah met by Gerkens before flicking off Simunovic and past Gordon.
Simunovic almost made amends immediately, the ball falling to the defender in the penalty area, and he turned sharply and fired just wide on the half-volley.
Anderlecht continued to press with Gordon saving well from Sofiane Hanni as Brendan Rodgers' men did just enough.
Key Opta stats:
- Celtic are on a run of seven Champions League home matches without a win (D1 L6) – this is their worst run in the competition at Celtic Park.
- Celtic have failed to keep a clean sheet in their last 12 Champions League home games.
- This was Anderlecht’s first away victory in the Champions League since December 2005, when they beat Real Betis (1-0) and their first win in the competition this campaign having lost all of their previous five games.
- Anderlecht have kept a clean sheet in all of their previous eight victories in the competition.
- Brendan Rodgers has won only two of his 18 group games as manager in the Champions League, with Liverpool against Ludogorets in September 2014 (2-1) and with Celtic against Anderlecht in September 2017 (3-0).