A Professional Game Match Officials Limited statement on Saturday evening revealed the apology after Marriner dismissed Gibbs for a foul committed by Alex Oxlade-Chamberlain which resulted in a penalty in the 17th minute of the match.
"Andre is an experienced referee and is obviously disappointed that an error of mistaken identity was made in this case," the statement read.
"Incidents of mistaken identity are very rare and are often the result of a number of different technical factors.
"Whilst this was a difficult decision Andre is disappointed that he failed to identify the correct player.
"He expressed his disappointment to Arsenal when he was made aware of the issue."
Marriner was aware of his error at half-time, but by then the decision could not be undone and marred Arsene Wenger's 1,000th match in charge of Arsenal.
With the Gunners already trailing 2-0, Gibbs was sent off for handling an Eden Hazard shot, despite replays clearly showing it was team-mate Oxlade-Chamberlain, who then approached Marriner to admit wrongdoing.
The Football Association subsequently confirmed that Arsenal, or the governing body themselves, could appeal the sending off on the grounds of "mistaken identity".
It continued the debate for the use of television replays.
The decision would not have happened under UEFA's extra-officials policy, according to a spokesman for president Michel Platini.
Pedro Pinto, the chief of press for Platini, said an extra referee on the by-line would have identified the correct culprit.
Pinto wrote on Twitter: "What confusion at Chelsea-Arsenal with the pen decision. That's why over 30 European countries have endorsed UEFA-backed 5 officials system.
"With an additional assistant referee on the end line, referee would not have got that sending off wrong. Technology is not the answer..."
Pinto claimed that "more eye balls" were the key to improving referee decision-making rather than technology alone.
The Barclays Premier League has employed goal-line technology for the first time this season - which would not have aided Marriner's decision at Chelsea because it determines whether a shot has crossed the line only.
Pinto added on Twitter: "More eye balls are the answer. GLT helps with goal line decisions, but five officials system gives referee more angles of vision. It's not one vs the other."
The only saving grace on this occasion was that the decision did not appear to impact on the result and it is likely that Oxlade-Chamberlain will serve the suspension if Arsenal appeal or the Football Association intervene.
"Clubs can appeal mistaken identity to The @FA, although The @FA can pro-actively review also," an FA tweet read.
There is a precedent for the decision to be changed after League One side Preston had the wrong man sent off against Port Vale in November.
Striker Joe Garner was suspended retrospectively for three games after midfielder Neil Kilkenny had mistakenly been sent off.
Marriner has previous form at Stamford Bridge this season, awarding Chelsea a controversial last-gasp penalty against West Brom in November when it appeared Ramires dived.
Hazard scored on that occasion to earn a draw in a result which preserved Jose Mourinho's unbeaten Premier League record at home, which now stands at 76 matches.
Former Premier League official Dermot Gallagher later gave Marriner his support.
Speaking on talkSPORT he said of the decision: "It's quite easy, it's just a genuine error, the referee, for whatever reason when the penalty has been given, has lost sight of the Arsenal player, and when he has looked back he's seen Gibbs and sent him off."
Gallagher claims it is an easy mess to clean up, though, adding: "Arsenal will appeal, that's their right. Kieran Gibbs, he will have to submit saying 'it wasn't me', Oxlade-Chamberlain will say 'it was (me)' and the suspension will be transferred and justice will be done.
"It's a genuine mistaken identity and Oxlade-Chamberlain won't be able to play (in Arsenal's next game)."