Wanderers had not beaten, or even scored against the Latics, in four attempts before today's game.
But Owen Coyle's rejuvenated side took a huge step towards securing their Premier League survival with a first victory over Wigan since December 2007.
Bolton are now eight points clear of the drop-zone after winning three of their last four games.
Wigan, buoyed by Monday's victory against Liverpool which ended their seven-game winless run, felt they should have had a penalty in the fourth minute when Zat Knight appeared to bundle over Hugo Rodallega, but referee Mike Dean waved away the protests.
But from then on it was all Bolton. Elmander fired the home side into a tenth-minute lead when Gary Caldwell made the first of a catalogue of errors on a terrible afternoon for the Scottish defender, and the Swede slipped the ball under Chris Kirkland for his first league goal for Wanderers in 2010.
Charles N'Zogbia came close to equalising for the Latics two minutes before half-time but he was denied by a superb save from Jussi Jaaskelainen after the ball had dropped kindly to the French winger eight yards out.
Bolton made a dream start to the second period when, just a minute after the restart, Caldwell tripped Tamir Cohen inside the area and Davies sent Kirkland the wrong way from the penalty spot.
Muamba made sure of the points seven minutes later when he twisted one way and then the other before drilling a low shot into the far corner for his first goal for the club.
And substitute Taylor pounced on another blunder from the hapless Caldwell to complete the rout on 69 minutes.