The Toffees started strongly, with Victor Anichebe looking bright down the right and his early cross was almost headed home by Nikica Jelavic.
However, it didn’t take long for David Moyes’ men to break the deadlock and it was Anichebe again who provided the assist, as his cross was deflected home by the impressive Steven Pienaar on 18 minutes.
The goal seemed to spark the home side into life and they levelled the scoring through Marvin Sordell, who ghosted into the box to fire home and send the Reebok into rapture.
Both teams continued to push forward but neither could find a goal and they went in level at the interval.
The end-to-end action continued after the break and the Trotters were unlucky not to take the lead as Sordell again went close following good work from Chris Eagles.
However, Everton started to take control in the closing stages with Pienaar spurning a number of half chances.
It almost looked as if it wasn’t going to be the Merseysiders but their pressure eventually told as after a goalmouth scramble, Heitinga fired home from 12 yards in stoppage time to send the Toffees through.
Dave Artell highlighted Jordan Bowery's important role in Crewe's battle against relegation after the Railwaymen beat Forest Green 3-1.