Griffiths, a known Hibernian supporter, will be the subject of a hearing at Hampden on Thursday over video footage showing him singing in an Edinburgh pub about former Tynecastle player Rudi Skacel being a "refugee" and chanting about the Tynecastle club "going bust".
He has apologised for his conduct and is likely to receive a suspension, but the SFA's decision to handle the issue after the first round of fixtures following the 33-game split permitted the 23-year-old to play against Motherwell at Fir Park on Saturday and score in a 3-3 draw.
Stuart McCall, the Lanarkshire club's manager, has drawn attention to the fact Griffiths may now be banned from playing against the other teams involved in the top six of the SPFL Premiership table and Anier, Motherwell's Estonian forward, is of the view that justice should have been handed down with greater urgency.
When asked if it should have been dealt with immediately, he replied: "Yeah, of course. That would be good and fair for the other teams.
"I heard someone talking on Saturday that he could have been banned for the game, but he played and scored. It could be bad luck for us, but it's not really my type of thing to answer."
Anier also pointed to recent refereeing decisions that have conspired against his side as they endeavour to beat Aberdeen to second place. They lost 3-2 to St Mirren earlier this month, thanks to a late and highly controversial penalty decision given by Iain Brines against Stephen McManus for handball, and he also felt they should have been awarded a spot kick against Celtic with the scoreline at 2-2 when Georgios Samaras appeared to illegally control a cross from Zaine Francis-Angol inside the area.
McCall claimed after Saturday's match that it was almost inevitable Griffiths would find the net since replacing Beram Kayal on 66 minutes with the match level at 2-2.
"I knew Leigh would come on and score," said McCall. "Nothing against him. He's a good kid, but apparently he is up on this [SFA] fast track that has got slow. If he gets banned, he will probably miss three or four games against our rivals. He might get off - who knows? - but this was supposed to be fast-tracked."
Stefan Johansen, the Celtic midfielder, was prevented from talking about Griffiths' SFA hearing by a club media officer, but made it clear the members of the SPFL Premiership champions' dressing room were delighted to see the striker find the net and remain supportive of him despite his problems.
"It's good that he scored, but Leigh has done very well since he joined Celtic," said Johansen.
"He has a bit of pressure on him, but he always delivers. He's a fantastic footballer and we were all happy for him.
"If you see him in training every day, he looks sharp and he is a great goalscorer."