Daniel Ayala's header 22 minutes from time earned the win after George Friend had cancelled out Milan Djuric's ninth-minute opener.
The triumph moved Boro up to fifth after Tuesday's defeat at Sheffield United saw them drop out of the top six.
"First half at Sheffield United wasn't good enough, the application and desire was very poor, second half was fantastic," Pulis said.
"We've spoken about it, not getting uptight and that's what the players have to do, embrace the challenge.
"They're fortunate lads, these are great games to play in and manage and that's the attitude they must have, roll your sleeves up, puff your chest out and work your socks off when you don't have the ball. We did that today."
Middlesbrough now face play-off rivals Derby and Millwall in succession and they will have to wait to see if Patrick Bamford will be fit.
There was 11 minutes of added time at the end of the first half after Bamford was laid out on the floor after a challenge from Aden Flint.
Pulis said: "Patrick's in the dressing room now taking some stick off the players so he's fine. He went down either winded or he's hit his head but he's fine.
"I've not spoken to him or the physios, I've just hammered him for being soft. Only joking. I have hammered him though!"
Bristol City have dropped four points off the play-off pace but manager Lee Johnson believes his players can still get there.
Johnson said: "The boys worked hard, effort and energy was good but they had more quality than us particularly in midfield with the decisions they made.
"We've conceded two poor goals, we talked about being good in flow, but also in the clutch phase dealing with corners, set-pieces but that's where we failed.
"We've had a few decent chances, Bobby Reid had a decent opportunity to be clinical but we just couldn't find that second goal.
"We had Lloyd Kelly, a 19-year-old academy graduate, against the fastest player in world football (Adama Traore) so we had to give him plenty of help.
"We've got to give it a right go to win the last three games, you've seen we're capable of doing that.
"We've got three winnable games if we're at it, but we have to have as many players at eight or nine from 10 to do that. We're learning in flight and have to take the lessons of games like this."