The visitors went in front early on through Gerard Deulofeu, a lead that was doubled on 20 minutes through England international Ross Barkley.
Giorgio Rasulo pulled a goal back in some style just past the half-hour mark with a fine strike and the Dons went close to levelling through Joe Walsh and Darren Potter.
However, the Toffees made it 3-1 late on thanks to Muhamed Besic’s fortuitous header from Aaron Lennon’s right-wing cross.
Karl Robinson named an unchanged starting XI for the visit of Ronald Koeman’s side, however, they were behind inside two minutes when Deulofeu made the most of a defensive mix-up and eventually scored from close range.
Either side of a Dave Martin save from Kevin Mirallas, Daniel Powell hit the side-netting after a powerful run and Ben Reeves forced Maarten Stekelenburg into a stop at the near post.
Just as the Dons were coming into it, the Toffees went 2-0 up when the exciting Deulofeu used his electric pace to get down the right-hand side and cut the ball back for Barkley, who expertly found the bottom corner from 18 yards.
The Dons responded shortly afterwards as Ed Upson forced Stekelenburg into a smart save away to his right from a set-piece on the edge of the box.
After an injury to Ben Reeves, Rasulo was called into action earlier than expected midway through the first half and he made an immediate impact, shifting the ball on his right foot 18 yards out and smashing the ball home in off the crossbar. Some strike!
It was almost 2-2 not long before the break but Walsh and Potter were brilliantly denied in quick succession by Stekelenburg and then Bryan Oviedo – both clearances off the line.
There were chances at either end at the second start half – Nicky Maynard couldn’t quite reach a Powell shot which ricocheted towards him while Mirallas angled wide a cross from Deulofeu for the Toffees.
Among a flurry of substitutes for the visitors just past the hour mark, Brendan Galloway made his Stadium MK return – one of the 18 players to have progressed from the Club’s academy into the first team.
Aiden McGeady should have made it 3-1 on 72 minutes but Martin was alert to deny McGeady after Leighton Baines had got in down the left and cut the ball back.
Oran Jackson, fresh from signing his first professional deal, was introduced alongside scholar Brandon Thomas-Asante with just over 15 minutes remaining – the latter leading the line.
Muhamed Besic restored Everton’s two-goal advantage in the final five minutes when he headed home Aaron Lennon’s cross via a deflection of George Williams.
Overall, a good test for Robinson’s men against a strong top-flight outfit.
MK Dons: Martin; Williams, Walsh, Downing (Kasumu 84), Lewington (c) (Tilney 66); Upson (Jackson 73), Potter; Carruthers (Furlong 84), Reeves (Rasulo 22), Powell; Maynard (Thomas-Asante 73).
Unused subs: Burns, Osei-Bonsu, Church, Tshimanga.
Goals: Rasulo (31).