Aitor Karanka hailed an "amazing" performance and felt Petr Cech was Arsenal's best player in Saturday's 0-0 draw at Emirates Stadium.
Boro remain fourth bottom and winless since beating basement boys Sunderland in August, but six of their seven points to date have now come away from the Riverside Stadium.
Adama Traore and Gaston Ramirez both drew superb saves from Cech, while the latter rattled the angle of post and bar from a first-half free-kick.
Mesut Ozil had an injury-time goal ruled out for offside for the hosts but Arsene Wenger's side struggled for clear-cut chances and their usual fluency.
"I think it was an amazing performance from my team because, especially after the last defeat at home [to Watford] it was tough for us," Karanka said.
"The plan from the training ground worked well for me and for us. It's really good when you are coming the Emirates and the best payer of Arsenal is Petr Cech.
"We showed everyone we could compete against the best teams in the Premier League.
"We knew we would have to work hard without the ball. They have a lot of quality players and you have to defend. We were really, really organised. Our transitions were really good and I'm really pleased.
"This team and this crowd deserves to live these kind of moments.
"In the Premier League every game is important. We can enjoy it for a few hours but tomorrow I will start to work for [next weekend's match against] Bournemouth. We can't stop if it is just one point.
"We have to grow, we have to learn from the experience. The best thing to grow and to add to that experience is by adding points, showing everyone we can compete against every single team."
Former Barcelona youngster Traore made his first start for Middlesbrough and the winger's electric pace posed regular questions of Arsenal's backline.
"He is a player who can make an impact. He is not here just to make an impact [from the bench], he is here to play from the beginning," Karanka added.
"He is young, he needs to learn. It has been a tough week working with him on the video and on the training ground but he is learning quick and he is a good player for the future for us."
