While the Briton finished outside the top ten, teammate Carlos Sainz came home in sixth place, matching the team's best result this year, as Norris picked up a P6 finish in Bahrain.
Sainz was then able to re-join the track after his pit stop in clean air, staying ahead of the Toro Rossos.
"I think I could have finished 10th at best but nothing more, and because of that we decided to try and get more points for Carlos and do better as a team," Norris said.
"They told me what I needed to focus on laptime-wise and obviously I was trying to look after my tyres at the same time, which cost a bit of pace.
"They told me what I needed to do and that allowed him [Sainz] to box cleanly ahead of me and, yeah, if they hadn't have done that, we probably would have been passed by the Toro Rossos because they were quite a bit quicker.
"I think they did a very good job and we did our homework before the race because we thought this might happen and we were well-prepared, and thankfully that got us some good points."
McLaren's strong result sees it pull away in fourth in the championship, as it now holds a 13 point gap to Racing Point.
Norris insists that he was happy to be a team player and sacrifice his own race, as he was aware that scoring points after a poor start was unlikely.
"I knew I wasn't going to do anything special or get loads of points so I just did the job I had to do," he said.
"As a team, we know what we have to do to get the best points possible and for the future in terms of constructors points and money for next year and improving as a team. Today was the best job we could have done so I'm very happy with the job that everyone did."