The Lions had settled for a point in their last four Championship games, and six of their previous nine in the league this season, but they were able to celebrate victory at last thanks to Jed Wallace's penalty.
After what was just their second league win of the campaign, Rowett said: "We all felt disappointed with the draws.
"I can assure that in the last couple of games, to come in with one point felt like we'd lost the game.
"The players, you could see it, felt like they'd lost the game because everyone was frustrated, so we just needed that tonight.
"I think we needed that lift, I think we needed that three points - it didn't really matter how we got it.
"I thought there were moments where we played quite well, moments where we looked a bit edgy, but the three points certainly makes our run look a bit better.
"It gives us a chance of one more game before the international break - it'll be a tough game [at Barnsley], but if we can go and put another big performance in, you see how the league table changes after three points at this moment in time."
Benik Afobe was denied by a fine save by Daniel Bentley after being put through by George Evans before the Bristol City goalkeeper thwarted him again before half-time from a header.
If was Afobe who won the penalty after 63 minutes, when he was bundled over by Rob Atkinson, and Wallace stepped up to hammer home from 12 yards.
Assistant manager Curtis Fleming took charge of Bristol City for the first time since Nigel Pearson fell ill with Covid-19 and he gave an update on how his boss was faring.
"We had a couple of messages during the game and Nige is OK - the Covid has hit him quite hard," Fleming said.
"If he had any chance of being here, he would have been and he's not feeling great.
"That's what I said to the guys after; if you're feeling a bit hard on yourself or a bit down, there's someone lying in Bristol very ill, that's when you have a problem."
On the defeat, Fleming added: "It was an attritional Championship game, which I thought came down to fine margins.
"We had a lot of possession and a lot of chances around the edge of the box to create a bit more and I don't think we did enough.
"I think that he [Afobe] may have looked a little bit offside, but it's the fine margins of the defensive third."