Emery was seen rushing back to the dressing room after the final whistle, while Arteta appeared to be waiting at the touchline.
"After the match, always my routine is quick - shake hands and go with my coaches, with my players, or go to the dressing room," Emery told reporters when asked about his quick exit.
"And I was waiting, and waiting.
"Of course, he was happy and he was with his coaches and I decided to go inside.
"But for me, it's not a problem," added Emery, who spent 18 months in the Arsenal hotseat before he was sacked in November 2019.
Arteta also played down the significance of the missed handshake. "No, that's OK. That's part of the game, not an issue at all."
Villa, in third, now trail leaders Arsenal by six points and next host Nottingham Forest on Saturday.