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.
Potters 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 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.
Everton Santos' first-half goal was enough for Seongnam to sink Sunderland 1-0 in their opening pre-season match in the Peace Cup.
The K-League side were given the lead on the half-hour mark when the Brazilian fired home from close range, beating Black Cats keeper Simon Mignolet.
Martin O'Neill's side pushed for an equaliser late on but Connor Wickham's deflected effort hit the post and Craig Gardner came close with a 30-yard free-kick which struck the bar.
Championship side Ipswich drew 1-1 away at Den Bosch in the second game of their tour of Holland.
After a 1-1 draw with Helmond Sport on Tuesday, Aaron Cresswell's late free-kick ensured a repeat scoreline against their Eerste Divisie opponents, cancelling out Randy Wolter's spectacular 25-yard strike.
Menawhile, League One champions Charlton grabbed a memorable 1-0 victory away at Portuguese giants Sporting Lisbon.
Johnnie Jackson's header in injury time gave the Addicks the win at the Jose Alvalade stadium, their second pre-season win of the summer.