Heskey won the free-kick, tempting Khalid Boulahrouz into a rash challenge, and the former Liverpool striker then got to Josip Skoko's set-piece delivery before Drogba.
Liverpool eventually broke free of their second half shackles and Wigan had Josip Skoko to thank when he threw himself at a Dirk Kuyt shot from 12 yards with Kirkland committed.
Skoko swung a free-kick into the box and, with no Wigan players attacking the ball, Weaver came out but failed to collect as Dunne glanced a header into the empty net.