Griffiths’ 38th minute opener, an own goal from Senegalese defender Serigne Mbodji early in the second half and a stoppage time strike by Scott Sinclair gave the Scottish champions a fully merited three points against a spirited but limited Anderlecht outfit.
Rodgers is now unbeaten in his last three away group stage games as Celtic manager, bucking the dismal trend on their travels which has handicapped so many of their previous campaigns.
Reaching the last 16 of the tournament this season remains a long shot but in the battle for third place and progression to the Europa League knockout stages after Christmas, the significance of this result cannot be overstated.
Before the outstanding piece of football which created the breakthrough for Griffiths, there were lengthy spells when the action lived up to the widely accepted pre-match perception of two teams scrapping for the crumbs from a Group B table lorded over by Paris Saint-Germain and Bayern Munich.
Olivier Ntcham, restored in central midfield at the expense of Stuart Armstrong, would justify his selection with the perceptive and magnificently weighted pass which led to Griffiths’ personal landmark strike.
But it took the French midfielder time to find his rhythm in a scrappy contest which saw him initially as guilty as anyone of losing possession cheaply. Indeed, Ntcham was fortunate to escape an early caution when he cynically fouled Lukasz Teodorcyzk in his attempt to win the ball back after giving it away needlessly to Pieter Gerkens.
Ntcham was penalised again when he floored Anderlecht captain Sofiane Hanni, conceding a free-kick in a dangerous position, but Leander Dendoncker’s effort from the set piece flew over without troubling Craig Gordon.
Celtic were seeing as much of the ball as they wanted, with the home side apparently quite happy to wait for opportunities to spring rapid counter attacks. The visitors looked vulnerable when Hanni’s ball over the top of their back four found Henry Onyekuru sprinting into the penalty area but he was unable to latch onto it as Gordon raced out to cover the threat.
Rodgers’ men were struggling to bring any penetration to their play in the attacking third of the pitch but Sinclair finally found space to force their first corner of the evening in the 25th minute. The winger took it himself, the ball nodded back across goal by Griffiths for Jozo Simunovic who was pulled up for a foul on Uros Spajic.
It was the beginning of the end of Spajic’s night, the Serbian defender then aggravating an ankle injury when fouled by Sinclair and eventually being replaced by Mbodji before the interval.
He hung around long enough to be on the pitch when Celtic took the lead with seven minutes of the first half remaining. Ntcham more than compensated for his earlier fecklessness when he conjured up a magnificent defence-splitting pass to send Kieran Tierney scampering down the left.
The full-back had all the time he needed to look up and drill the ball low across the face of the Anderlecht six-yard box for Griffiths to drill in a shot from close range at the far post. Moussa Dembele, an unused substitute on the night, has a battle on his hands to reclaim pole position up front in Rodgers’ team.
Celtic were almost caught cold from the restart, allowing Hanni to work himself into space on the edge of their penalty area. The Algerian’s powerful low shot was well gathered by the alert Gordon.
It was a quick reminder that Celtic still had plenty of work left to do in order to achieve the optimum outcome from this pivotal fixture. Anderlecht threatened again on the stroke of half-time, Teodorcyzk finding room in behind the right side of the Celtic defence before dragging his shot across the face of goal and narrowly wide of the target.
Anderlecht opened the second half on the front foot, Hanni’s fierce shot from 20 yards blocked by the chest of Gordon, but their hopes of staging a recovery were dented when Celtic doubled their lead in the 50th minute.
There was an element of good fortune to the goal from Celtic’s perspective as Anderlecht gave the ball away on the edge of their own penalty area. Patrick Roberts seized upon the error, cutting inside from the right and firing in a shot which Mbodji could only deflect beyond his own ‘keeper.
It was now just a case of managing the game sensibly for Celtic, a task disrupted a little by the departure of captain Scott Brown with an injury which will be of as much concern to Scotland manager Gordon Strachan as it is to Rodgers. Nir Bitton replaced Brown as Celtic looked to navigate the closing stages with as little anxiety as possible.
A thunderous 25 yard shot from Gerkens, brilliantly touched over by Gordon, was the closest Anderlecht came to pulling a goal back as Celtic saw it out with something to spare, adding gloss to the final scoreline when Sinclair collected a Ntcham pass and lashed the ball high beyond Frank Boeckx in stoppage time.