Substitute Stevie May netted the only goal of the game, returning to haunt his former club with a low strike from the edge of the area that looked like it should have been saved by goalkeeper Joe Lewis.
But Davidson was more concerned with the result, saying: "It's great. If you asked me what was our worst performance in the league, you'd probably say that was it. We kept at it.
"I thought defensively we were really good. Our shape and energy were good. We didn't control the game enough going forward first half but we grew into the game and had a couple of really good chances towards the end.
"Efe Ambrose is a great addition to our defence, especially with Liam (Gordon) injured and Jason (Kerr) away. I also thought James Brown was outstanding. It's only the second time he's played as a centre half and he was really good defensively.
"Neither side really had the quality in the final third but I've been on the other end of that this season so I'm delighted with a positive result."
Aberdeen boss Stephen Glass was left disappointed and frustrated by his side's performance.
"We didn't have the quality in the final third and defensively we weren't good enough when the game was on the line," he said. "They've got situations they can get shots off and it needs to be better in defence.
"You're looking at two teams not doing what they should do and players are feeling it. There's a lot of quality in both teams, but they're trying to stop each other. We created a decent amount but the quality in the final third was lacking.
"I'm disappointed, the players are disappointed and the fans go home frustrated. The intent is there from the team and I think it will turn. You have to keep pushing and keep working. There's no reason for us to doubt the group of players that are here."