The 34-year-old returned to the West Brom side for the first time in over two months and scored a brace in an entertaining Barclays Premier League game in which both sides came from behind.
Anelka seemingly performed the 'quenelle' salute, made famous by French comedian Dieudonne and linked to anti-Semitism in his homeland, but Downing said the striker was merely paying homage to his friend.
Anelka scored twice in five minutes at the end of the first half after Joe Cole had put West Ham ahead but it was his opening strike, which was his maiden West Brom goal, which led to the celebration.
Downing said the 'quenelle' gesture was in no way linked to anything other than a dedication to Dieudonne, a personal friend, and said anything else being read in to the display was "absolute rubbish".
"I'm aware of it but it has got nothing to do what is being said," Downing replied when asked about the gesture.
"It is dedicated to a French comedian he knows very, very well. He uses it in his act and I think speculation can be stopped now, it is absolute rubbish really.
"He is totally unaware of what the problems were or the speculation that has been thrown around, he is totally surprised by it."
Press Association Sport understands the Football Association is aware of the allegation over Anelka's celebration and is looking into the matter.
Anelka had not played since the 4-1 defeat to Liverpool on October 26 but was recalled as one of six changes as Downing looked to prevent any tiredness with the game coming less than 48 hours after a Boxing Day draw at Tottenham.
The caretaker manager was pleased with Anelka's display and is adamant the former Arsenal and Chelsea man is happy at the Hawthorns.
"I thought he had a good game," Downing added.
"He combined with [Saido] Berahino really well. He has the quality to drop in and link the play and controls the game from that area.
"He got on the shoulder for his first goal and then a typical poacher's goal from the corner.
"It has been a while since he has played but his dedication and attitude has always been spot on. He is prepared to play and wants to play."
West Ham did not look like getting back in to the game until a manic four-minute spell in the second half yielded three goals.
Half-time substitute Modibo Maiga drew the Hammers level with a curling finish before setting up Kevin Nolan two minutes later.
Berahino was on hand to peg West Ham back once more as both sides had to settle for a point.
It means Sam Allardyce's side remain in the Premier League bottom three and the 59-year-old was left bemoaning his extensive injury list, which had James Tomkins' name added to it when the defender was forced off with a groin problem after just 10 minutes.
"We didn't have any problems apart from James Tomkins getting injured," Allardyce said about his side's start to the game.
"It was a remarkable performance from our players considering we had four full-backs playing across the back four today.
"Had we had a full back four fit today, never mind all the other injuries, I think we would have won. Unfortunately, with players playing out of position we have conceded three goals."