Teenage midfielder Ross Barkley was the hero for the Toffees at Tannadice, nodding home Seamus Coleman's cross with three minutes remaining.
That makes it two wins from two in pre-season for David Moyes' men, following their 4-1 victory at Morecambe on Saturday.
Stoke kicked-off their preparations for the new campaign with a 1-0 victory away at Swiss side FC Sion.
Winger Matthew Etherington was the man on the scoresheet, converting from the penalty spot in the 38th minute after defender Jose Goncalves had handled in the area.
New West Brom boss Steve Clarke suffered a 1-0 defeat in his first game in charge of the Baggies against Malmo.
Alex Nilsson's eighth minute strike was good enough for the Swedish outfit in the opening game of West Brom's Scandinavian tour.