Saturday's clash at St Mary's was teed up as make or break game for Roberto Mancini's side as they looked to retain the Premier League crown. City, though, rarely resembled a side in such dire need of victory, putting in what Mancini described as a "worse than poor" performance as Southampton secured a 3-1 success.
However, Davis was just pleased to secure a first win since Pochettino replaced Nigel Adkins as manager, moving Southampton six points clear of the drop zone in the process. "We're obviously delighted with the result," Davis said. "We knew we would obviously have to put a lot of pressure on them up the pitch."
He added: "That is the way the manager wants us to play and we managed to do that. We caused them a few problems and deserved the three points. It is hard to look at other teams at this point in time. You've just got to concentrate on yourself.
"We've shown that if we can keep the consistency going and level of performance we are more than capable of staying in this division and hopefully kicking on."
Davis scored Southampton's second goal on Saturday, making the most of a Joe Hart howler to extend the lead given to them by Jason Puncheon. Edin Dzeko pulled one back for City before half-time but Saints rallied, with Gareth Barry putting into his own goal to make it 3-1.
The only thing missing from a perfect Saints victory was a 100th goal for the club from Rickie Lambert, who put in another impressive display despite failing to score.
"He is a great 'out ball' for us," Davis said. "Obviously he has got a lot more to his game than just holding up the play, but that is an important aspect of his play as well. He brings other into the game and allows others to get up the pitch. He did that really well."
Unsurprisingly, City's players were less chatty than their Saints counterparts after the match. Only manager Mancini spoke to the media at St Mary's, although some players took to Twitter in the aftermath.
"Some days, nothing works the way it should," Argentina international Sergio Aguero said via his account. "This was one of those days. There's no excuse - we need to work hard to forge ahead. It all looks a bit grim right now, but I won't believe giving up is an option."