Wayne Rooney had put the Toffees into an early lead, but Arsenal dominated for large periods as Mesut Ozil marked his return to the team with a goal and an assist to help compound Everton to yet another defeat.
Ozil and Alexis Sanchez had both been recalled to the Arsenal starting lineup and it took the pair less than three minutes to make a contribution, with their work leading to Aaron Ramsey having a shot superbly tipped past the far post by Jordan Pickford.
Ramsey soon turned provider for Alexandre Lacazette, who was denied by Pickford from close range, but after being second best during the opening stages, Everton went ahead.
After Idrissa Gueye dispossessed Granit Xhaka, the ball fell to Rooney who curled the ball into the far corner from 20 yards out - almost 15 years to the day since the former England skipper announced his arrival in English football with a goal of similar quality against the Gunners.
Despite going behind, Arsenal remained in the ascendancy and Pickford had to be alert to keep out Ramsey after fumbling Sanchez's 30-yard effort from a free kick.
Back up the other end, Dominic Calvert-Lewin flicked a header wide of the far post, but the rest of the opening 45 minutes belonged to the visitors who eventually equalised just before the break.
Within eight minutes of the restart, Arsenal were deservedly ahead thanks to Ozil, who timed his run to perfection in order to latch onto Sanchez's cross and the German's header from eight yards left Pickford with no chance.
Shortly afterwards, Ramsey blazed over from just 12 yards out after being found by Laurent Koscielny's pull-back, and it left Koeman with decisions to make as the Toffees continued to look open at the back.
Koeman remained patient before making further alterations after already introducing Tom Davies at half time, but the Dutchman's hand was forced when Gueye was dismissed for a second yellow card after a wild lunge on Xhaka.
Six minutes later, Arsenal made sure of all three points when Ozil was given time and space on the right-hand side of the penalty area before picking out Lacazette, who had the simple task of firing into the corner from eight yards out.
With a man advantage, Arsenal were able to control the closing stages and the visitors deservedly added a late fourth thanks to Ramsey, who collected Jack Wilshere's pass before slotting past Pickford from inside the penalty area.
The goal sparked a late flurry of action in both penalty areas, with Niasse capitalising on a mix-up between Petr Cech and Monreal before walking the ball into the net.
However, the last action of the game belonged to Sanchez, who danced around three Everton defenders before lashing the ball into the far corner of the net to complete a fine afternoon for the visitors, who now sit in fifth place in the standings.