After goalkeeper Maarten Stekelenburg's controversial own goal, Gareth Barry, on his 600th Premier League appearance, equalised before Seamus Coleman and Romelu Lukaku - who later went off with an injury to his big toe - turned things around before half-time.
"Maybe we needed a yes and no mistake of the referee as after that we showed really good football, were more aggressive," Koeman said.
"I think Middlesbrough started better but there was great character of the team being 1-0 down at home and it was 3-1 by half-time.
"It was really enjoyable how we played."
Koeman praised the longevity of Barry after another commanding performance from the 35-year-old.
"It is amazing the number of games Gareth has played and he showed he was a perfect player for the team," said the Toffees boss.