The Spain striker provoked a red card for Arsenal defender Gabriel late in the first half with Santi Cazorla also seeing red in the second period as the visitors self-destructed.
Kurt Zouma headed Jose Mourinho’s men into the lead after 53 minutes and a late, deflected effort from Eden Hazard secured a much-needed win for Chelsea in stoppage time.
Gabriel became the latest man to be drawn into an off-the-ball row with Costa, who had first tangled with Laurent Koscielny but was involved in afters with the Frenchman’s defensive partner.
Gabriel received an initial booking from referee Mike Dean before his sending-off for a petulant kick on the Spain international before play had resumed on the stroke of half-time and it took the champions just eight second-half minutes to make the most of their man advantage.
The unmarked Zouma nodded in a Cesc Fabregas free-kick past Petr Cech on the goalkeeper’s Stamford Bridge return.
Chelsea found their feet after the opener and rarely looked like relinquishing their lead – with Cazorla’s second yellow card for a poor tackle on Fabregas effectively ending a fixture unlikely to be earmarked as a classic episode in the sides’ long-running rivalry.
Hazard’s goal merely rubbed salt into Arsenal’s wound and summed up the visitors’ luckless afternoon.
John Terry was kept on the bench following the midweek Champions League success against Maccabi Tel-Aviv and stand-in captain Branislav Ivanovic was tested early down the left by Alexis Sanchez before Theo Walcott rounded Asmir Begovic before failing to keep the ball in play.
Former Arsenal man Fabregas had Chelsea’s first effort on target with the opening 15 minutes largely belonging to the visitors – unchanged from their 2-0 win over Stoke City.
Mourinho’s men eventually grew into the game as Costa tested Cech with a strike prior to the half-hour mark.
A slack touch from Pedro denied him a one-on-one with Cech before Hazard saw a penalty shout turned down after tangling with Gabriel – Mourinho seemingly bemused by Dean’s decision.
Dean was then involved in the game’s main talking point prior to the break with Gabriel dismissed to the astonishment of Wenger.
Wenger introduced Calum Chambers to shore up his defence but Chelsea remained on top as they sought to press home their numerical advantage, Pedro volleying over before Zouma nodded in the opener.
Ghosting in at the far post, he gained the crucial touch on Fabregas’ delivery to register his first Premier League goal for the club.
The hosts’ pressure continued as Hazard’s fierce effort was beaten away by Cech, with the former Chelsea goalkeeper then watching on as the Belgian dragged a shot wide after the hour mark.
Costa remained in the thick of things – seeing a penalty appeal turned away following a challenge from Nacho Monreal – with Cazorla shown a second yellow for a rash challenge on his countryman Fabregas 11 minutes from time before Hazard sealed the points.
His strike took a heavy deflection off substitute Chambers and diverted past a helpless Cech to cap another frustrating visit to west London for Wenger.