A solitary goal from Craig Davies was enough to give Barnsley the win, and Hill believes that his team's performance deserved all three points.
"The performance merited the result. We had opportunities to score more goals on the counter-attack," said Hill.
"I am over the moon with the players' efforts and their pace. I thought our keeper (David Gonzalez) was excellent on his debut. I would have been disappointed if we had lost that game.
"I expected Middlesbrough to give a bit more than they did. I think they are short of a target man, but they moved the ball around in the right areas. I still believe that they one of the favourites to get a playoff place."
Hill reserved special praise for Davies who, in addition to scoring the winner, came close with a number of other efforts.
"Craig needs to play off someone and if he does that this season he will score goals. Last season he was a target man and he's not a target man, and last season he was a manager's nightmare," added the Barnsley boss.
"Perkins made the right decision to play the ball to Davies because he had the opportunity to score. Davies made the keeper work with all his efforts. The keeper made some excellent saves from him to deny him a hat-trick."
Barnsley's next match comes against Wolverhampton Wanderers on Tuesday evening.