The 21-year-old Spanish winger scored the opening goal at the Riverside Stadium with an impressive individual effort before providing a lovely cross for Romelu Lukaku for the second after seeing off Fernando Amorebieta on the right flank.
Deulofeu's overall performance was the key to Everton's triumph and his display was certainly not lost on Toffees boss Roberto Martinez, as well as Boro boss Aitor Karanka, who coached him at youth level in the Spain national team setup.
"I see that from Gerard every day," said Martinez.
"I'm impressed with his maturity and the consistency he is showing, which is difficult as a winger."
Karanka added: "I have known Deulofeu since he was 14.
"When he is on his day, as he was today, it is difficult to stop him."