Forward Leigh Griffiths netted in the 38th minute after an initially slow first half - scoring after a lengthy series of 28 passes between Celtic players that ended with Kieran Tierney whipping in a flawless cross for Griffiths to strike home.
Now, pundit Ian Wright has told BBC Sport that whilst many will belittle Celtic, their result in Belgium was "very impressive", and manager Brendan Rodgers deserves major praise for the way he gets his side to raise their game in Europe after coasting through Scottish football.
Patrick Roberts then forced an own goal within five minutes of the restart and the Scottish champion capably dealt with Anderlecht's long-range attempts on its way to a maiden UEFA Champions League clean sheet away from home.
In the aftermath, Brendan Rodgers was happy to highlight the value of only a second Celtic away Champions League win in 28 games.
In a statement on Monday, Anderlecht said that they "would like to remind all fans of Celtic FC who will be making the trip to Brussels that it is mandatory to be in possession of a valid match ticket when they are present in the territory of Anderlecht or the neigbouring community of Sint-Gillis/Saint-Gilles".
"They will play better".
"Defensively we were exceptional in terms of our pressing and trying to get the ball back", Rodgers continued.
"It takes experience and time and it takes belief".
"I think people recognise this is a good side that is developing well".
"In spells with the ball we were very, very good".
"They can go in to those Munich games and enjoy them", he added.
Last term, Celtic managed just three points and were without a win during their Champions League campaign.
They will need to when they face Bayern Munich in a few weeks time.