Man United ultimately collected all the points at the Riverside Stadium with a 3-1 win, but Middlesbrough threatened an equaliser late on after Rudy Gestede had made it 2-1 with more than 10 minutes left to play.
Antonio Valencia's stoppage-time goal secured all the points for Jose Mourinho's side, but Agnew, in his first match in charge since Aitor Karanka departed, believes that Boro were good enough for at least a point.
"We're disappointed to lose the game - the second goal set us back a little bit," Agnew told reporters. "But in terms of the players, I couldn't be more proud of their fighting spirit.
"If we show that commitment and intensity until the end of the season, we have a real chance of staying up. It is a relief to get a goal, but we didn't quite get a bounce of the ball and the equaliser. On the balance of the game at that time, I thought we deserved it."
Agnew is expected to keep his role as manager until at least the end of the season.