Luke Chambers fired the only goal two minutes into the second half as Forest maintained their promotion challenge by leapfrogging Leeds into fifth place in the table.
"Efficient would be the best way of describing our performance,'' said Davies. "It was always going to be a tough game after the Derby game at the weekend, which was draining both mentally and physically. The maturity of this side is getting better, the discipline of the team is improving and it's a good clean sheet and three points.''
He added: "We knew Bristol would come and make it tough which they did. They came with two banks of four that sat in and they asked questions of us.
"We had to be patient and knew they would be a threat on the counter attack with the pace of Jamal Campbell-Ryce and Albert Adomah and then Ivan Sproule who came on late on. They are dangerous players.
"But it's a fantastic three points. I don't think we would have won that game last season and that's a credit to the players."
City boss Keith Millen saw his side slip to a fifth game without a win in all competitions and felt the performance summed up their efforts in recent weeks.
"We didn't have enough quality about our play in the final third of the pitch to trouble Forest as we might,'' Millen said.
"I thought that we had enough chances to get at them, even in the first half when we had a lot of opportunities on the counter attack.
"In the second half we had more control of the ball but more often than not our final ball or decision-making let us down. It was poor and that's the way it has been for the last few matches.''