Goals from Daniel Powell and Ryan Lowe - either side of Franck Moussa's equaliser - had put the Dons in charge at half-time, but Elliott's double condemned the hosts to a third defeat in four games.
Milton Keynes had the better of an exciting first half and Powell put them ahead after 15 minutes when he cut in from the right and drilled a fine left-footed shot inside the far post past goalkeeper Joe Murphy.
Moussa produced a superb solo goal to equalise for Coventry in the 27th minute, beating two Dons defenders before lashing the ball in off the underside of the crossbar.
Lowe restored the Dons' lead seven minutes before the break when he turned Sky Blues centre-back William Edjenguele inside-out before drilling a fine low finish across Murphy for his fifth goal in as many games.
But City striker Elliott scored twice in three minutes shortly after half-time to turn the match on its head, first when he nodded in Gary McSheffrey's cross from close range and then when he skinned Antony Kay to fire low past David Martin.