Wright-Phillips hit the second after Anelka had given Kevin Kilbane the slip to set up the shot. But Wigan went out with all guns blazing as substitute Antoine Sibierski pulled one back three minutes from time.
Chelsea had quickly shown their attacking intent hitting Wigan with a terrific move which should have produced a goal inside the first minute.
It ended with Joe Cole having a clear sight of the target but the England star hit it wide.
The Latics were looking vulnerable to the speed of the Chelsea counter-attack and had a let-off when Wright-Phillips mis-hit his cross after getting the better of Wilson Palacios with Anelka in the middle waiting to strike.
Wigan had been restricted to a long-range effort from Ryan Taylor which failed to trouble Petr Cech until Emile Heskey set up a great chance.
He played a superb ball to the feet of Marcus Bent but he was a split-second slow in reacting and Wayne Bridge got back to clear for a corner.
Wigan survived another nervous moment when Wright-Phillips did well to rob Taylor and set up the attack. But again Cole's first time shot flew well wide.
Fortune favoured Wigan again soon after the re-start when Anelka was pulled up for offside when he had a clear run at goal.
However, the French star was soon smiling, as he met Juliano Belletti's cross to prod over Kirkland as the keeper raced out.
It provoked fierce protests from the Wigan players, with Paul Scharner booked, but the replay suggested that the referee got it right.
Wright-Phillips was inches away from grabbing a second for Chelsea, but Wigan hit back with Heskey forcing Cech into action.
There were mass protests from the Chelsea bench - and an exchange of views between the two managers - after the Makelele-Brown incident.
The Wigan midfielder did catch him high, but there seemed to be no intent and no action was taken against him.
Wright-Phillips gave Wigan a mountain to climb but Sibierski gave them hope and, in stoppage time, Bent hit the bar with a terrific strike.