Ross Barkley's close range header sealed the win for the visitors in the dying stages after another solid test for the Blues.
Clear-cut chances had been a rarity as the teams huffed and puffed throughout a match lacking in excitement but rich in benefit to the players.
And while drama may have been in short supply, the Blues were given a thorough workout in the first trip of their mini-tour north of the border.
The match had been put back 24 hours due to a waterlogged pitch which gave Everton's stars an extra break after their 4-1 victory over Morecambe on Saturday.
Francisco Junior looked lively throughout but was taken off for Barkley after being booked for a lunge that would have been given as a red card had it been in the Premier League.
Barkley was immediatley involved in the action as his cross skimmed across a crowded penalty area.
The youngster bagged the winner through a Seamus Coleman delivery to round off a productive night north of the border for David Moyes' men.
Follow Gary Maiden on Twitter @GaryMaidenEFC