The Clarets had not won a game under Dyche in which they had conceded and so Burnley fans would have feared the worst after Albert Adomah put Bristol City in front on 35 minutes.
Despite dominating the first half Burnley had to wait until the 52nd minute to level through Jason Shackell, while late strikes from Chris McCann and Martin Paterson wrapped up the points. In between Burnley's second and third goals, Lee Grant produced a terrific penalty save to keep out Bristol City's Steven Davies.
"That was as dominant a display as I have seen from us this season," Dyche said. "We had 25 efforts on goal and I very much enjoyed our performance. We played with good ingenuity, organisation and freedom and even when we found ourselves behind at half-time we stayed calm.
"We always knew we could win the match. We made good use of the ball and managed the game well. All in all it was a very pleasing day for me, the team and the fans that backed us again and helped galvanise us as a group.
"I was really delighted with the way we went about our business but the madness of the Championship this season means that no-one can sit back and relax. Lee Grant made a fantastic penalty save but their goalkeeper made some top drawer saves too.
"After the penalty we went and scored again and that was a great way to respond. We have belief in what we do and we have seen that through the anxiety of recent weeks."
City boss Sean O'Driscoll did not appear for the post-match press conference, but goalscorer Adomah said: "I scored a goal and won a penalty for the team and if that had gone in the result could have been different and we might have come away with a point. We always back Steven to put them away but it just didn't happen for him because the keeper made a great save.
"If we had won then we could have kept up with the teams above us but it is not over yet and we have to stay positive because we still have five games to keep ourselves in this division. Everyone is trying to gee each other up and that encouragement is important because when we are down we don't want to be knocked again.
"We can still survive but we have been fighting ever since I came here and I try and do my best whenever I get onto the pitch. You can't dwell on things you just have to move on and hopefully we can do that as a team."