The Merseysiders were the better side in the early exchanges and went in front on 18 minutes when Apostolos Vellios crossed from the edge of the penalty area and John Lundstram cracked home.
Boro settled to become competitive but Jamie Clapham was forced into a change on 41 minutes when Matty Waters was replaced by Lewis Sirrell.
A smart save from Connor Ripley kept the Toffees' lead to just one when he tipped a Lundstram effort round the post in the early exchangesof the second half, but there was nothing he could do as Everton wrapped the game up midway through it.
Ross Barkley is known to a few of Boro's youngsters having worked his way through the England Youth ranks and he showed evidence of his international class on 66 minutes when slotting home from a narrow angle following a telling run down the left.
With 11 minutes remaining Marcos Rhoden, one the two Brazilians signed by Boro last week, had his first taste of English football when replacing Kyle Oliver.
Everton: Mucha, Stones, Garbutt, Junior, Duffy, Browning, Hope, Barkley, Velios (Duffus 79), McAleny, Lundstram. Subs: Springthorpe, Johns, McLaughlin, Adelson.
Middlesbrough: Ripley, Bennett, Weldon, Oliver (Rhoden 79), Burgess, Jackson, Waters (Sirrell 41), Morris, Fewster (Brobbel 66), Kneeshaw, Williams. Subs: Montgomery, Edwards.