A dour first half which gave both goalkeepers little to do preceded an action-packed second 45 minutes which saw three penalties awarded by referee Lee Mason but just one converted as David Moyes completed a snatch and grab mission at Ewood Park.
Early in the second half, Toffees youngster Ross Barkley lost possession in the Rovers attacking third and in the process of attempting to clean up his mess, tripped Mauro Formica with Mason having no choice but to point to the spot.
Junior Hoilett was the man with the responsibilities but he could not get his place kick past Tim Howard with the Toffees defence holding strong to deny the follow-ups.
The game opened up as time progressed but the next best chance was again a penalty when Phil Jagielka up-ended Formica in the area in the 83rd minute.
The Argentine, who had come on just after the half-hour-mark to replace the injured David Dunn, this time stepped up put rammed his shot into the post as the home side were again the victims of their own profligacy.
Spanish playmaker Mikel Arteta was the man to calmly slot home in injury time, sending Paul Robinson the other way as Everton made the great escape out of Lancashire with their first set of maximum points for the season.
It was a sad end for Rovers midfielder Brett Emerton who was playing his final match for the club after eight years of services as he gets set to return home to Australia after signing with Sydney FC but Socceroos teammate Tim Cahill and his Toffees men would be delighted to open their account as Blackburn fall to the bottom of the table with zero points from three outings.