Goodison Park had to wait until the 78th minute for the equaliser from James McFadden.
Rovers took the lead against the run of play early in the first half from Paraguayan international Roque Santa Cruz and looked on course for a second successive away win until the late leveller.
The game was preceded by a minute's silence for murdered 11-year-old Everton fan Rhys Jones, with his parents Melanie and Stephen and brother Owen standing alongside manager David Moyes near the dug-out.
"We did alright until they scored, but after that it was backs to the wall - we needed a little more craft.
"Hopefully in the next week or so we can strengthen the squad - but you never be confident until it's done."
Moyes also paid tribute to the minute's applause for murdered child Rhys Jones: "The reaction of the supporters shows what everyone feels. It's a travesty."