Victory for the Gunners left the Merseyside club in the Premier League relegation zone and proved to be the final straw for manager Ronald Koeman, who was sacked on Monday afternoon.
African-based betting company SportPesa, who agreed a "multi-year agreement" to be Everton's "main partner" in May, appeared to revel in Arsenal's victory with a tweet sent from its account which in part said: "Well in #Gunners! A superb gift by the players for the birthday gaffer! #MadeOfWinners."
The post has since been removed from the SportPesa account and it has apologised in a statement it addressed to Everton fans on Monday.
"We have to be honest in admitting that the tweet you say yesterday should never have gone our and we are deeply sorry for any offence this has caused," it wrote.
"We have reviewed and will revise our internal policies and processes, especially in relation to our other partner clubs, to ensure that this never happens again. We will up our game."
It is not the first time this season a betting company that has a sponsorship arrangement with a Premier League club has apologised for posts made on social media.
In September, Burnley's official UK betting partner Ladbrokes said sorry to the Clarets for "very poorly conceived tweets" made during their Carabao Cup contest with Leeds.
One was addressed to ex-Whites players Chris Wood and Charlie Taylor and featured a middle finger emoji.
Additional reporting by the Press Association.