Manchester City moved above opponents Southampton into second place as three second-half goals cut Chelsea's Premier League lead to six points.
Sergio Aguero, in his 100th top-flight game, pulled the ball back for Yaya Toure to drill in a low strike.
When the Premier League fixture list was released in the summer, few would have glanced at this match and marked it down as one between two teams potentially competing for Champions League qualification.
After a summer of upheaval, Koeman has rebuilt the south coast side on a strong foundation of defensive stability, with the Saints having conceded just six goals in their previous 12 top-flight matches.
But, after French holding midfielder Morgan Schneiderlin's injury forced a defensive reshuffle at half-time, they were picked apart by the visitors.
City striker Sergio Aguero, whose midweek hat-trick against Bayern Munich took him to 17 goals for this season, was expected to be the biggest threat to the home defence.
The Argentine did not add to his tally, instead turning provider for both Toure and Clichy, who poked in at the end for his first Blues goal.
However, Aguero should have been given the opportunity from the penalty spot on 10 minutes. The City talisman was scythed down by home skipper Jose Fonte's lunging tackle, but looked up to see referee Mike Jones showing a yellow card after the official decided he had dived.
But in the end, that controversial decision did not matter.
Many observers warned that Southampton's credentials would be tested in a difficult run against City, Arsenal and Manchester United - and in this game they looked short of the quality needed to break into the top four.
Koeman's men grew in confidence after a nervous start and could have opened the scoring when Steven Davis's low shot forced Joe Hart into a superb save low down to his right.
Speculative shots from Toby Alderweireld and Dusan Tadic flew past Joe Hart's right-hand post shortly after the restart, but once Toure scored his fifth goal of the season there appeared to be little way back for Saints.
And so it proved - despite Mangala, who was booked in the first-half for a cynical tackle on Tadic, seeing red after clipping home substitute Shane Long.
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