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Skipper Frank Lampard got the opener for the champions and Salomon Kalou got the second - his first goal in the Premiership.
But former England striker Emile Heskey sparked a spirited comeback by the Latics with an injury-time header and made it five for the season with the second-half equaliser before Robben broke Wigan hearts.
Wigan found their resources stretched to the limit with Henri Camara, Paul Scharner, Andy Webster and Antonio Valenica all on the casualty list and Lee McCulloch starting a three-match ban for punching Sheffield United's Chris Morgan.
It gave on-loan Bulgarian striker Svetoslav Todorov his first start, while Swede Andreas Johansson also got a rare chance, with fit-again Gary Teale on the bench.
Chelsea, still hot in pursuit of Manchester United, were again without skipper John Terry.
Ivory Coast star Kalou had two early chances to give Chelsea a flying start only to be foiled by keeper Chris Kirkland.
Robben set up the first opening, but Kalou only got the slightest of touches and Kirkland was perfectly positioned to hold it.
Didier Drogba and Robben then combined to give Kalou a clear sight of the target, but the shot lacked conviction and Kirkland turned it onto his left post and darted back to gather it at the second attempt.
But the champions continued to dominate and finally took the lead after 13 minutes. Dutch ace Robben slipped a great ball into the path of skipper Lampard and his shot went in through the legs of Kirkland.
There was a hint of offside, but replays suggested that Kevin Kilbane had played Lampard onside.
The Latics turned the tide briefly with Heskey heading over from an Emmerson Boyce cross, but Wigan were living dangerously at the other end as lively youngster Kalou fired across the face of goal.
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Hall thumped one over the bar as Wigan looked to get back into it, Kilbane forced an excellent save with a swerving shot from the edge of the area and Johansson also tested the keeper.
But Wigan grabbed a lifeline in stoppage time after Khalid Boulahrouz fouled Heskey just outside the area. Josip Skoko whipped in the free-kick and Heskey headed in his fourth of the season at the back post.
Lampard missed a great chance to restore Chelsea's two-goal advantage early in the second half, dragging a left-foot shot wide after Drogba set him up.
Wigan went agonisingly close when Johansson rattled one just wide, but they finally got their reward after 75 minutes.
Sub Dave Cotterill's cross was only half cleared and fell to Denny Landzaat who fired it back inside for Heskey to turn it past Hilario from five yards.
In a nail-biting finish, Kirkland thought that he had ensured that Wigan held on for a share of the points, turning a Ricardo Carvalho header over the bar, only for Robben to strike right at the death.