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 they won in such dramatic circumstances last season.
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.
That result swings the pendulum firmly in Manchester United's direction - not that Saints midfielder Davis is bothered.
The Northern Ireland captain 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.
"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 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."
Lambert was one of a number of man-of-the-match candidates, along with the likes of Jos Hooiveld, Jack Cork, Morgan Schneiderlin and Davis.
The latter has endured a difficult start to life at St Mary's after his summer move from Rangers, but believes he is now beginning to prove his worth.
"It has obviously been a little bit frustrating for me in terms of the fact perhaps I have not played as much as I would like," Davis said.
"I came down and missed a large chunk of pre-season and just came back from injury.
"I am feeling a lot fitter and stronger now and that I am getting back to my best. All I can do it try and make an impact on the manager."
Davis also explained his decision to leave his boyhood heroes while leaving the door open to the possibility of going back at some point in his career.
He told the Daily Record: "We were professional sportsmen and on the day it was announced Rangers were going into administration it was just a series of words to us until we absorbed the ramifications of what had taken place.
"I had just signed a new, five-year contract with the club and it had always been my intention to honour that agreement in full and devote the largest part of my career to the team I considered to be one of the biggest names in world football.
"I loved the club and the fans but I had to take a decision which was best for me, my family and the remainder of my career.
"There was so much uncertainty, so I was delighted Southampton made an offer that was acceptable and the club got money that they needed to keep going.
"It was the toughest decision of my life when I left for another club.
"I gave the team 100% every time I went out on to the park for Rangers.
"If everything goes to the plan I have in my head I'd like to come back and play for the club again before my career ends.
"I know there are different opinions about the players who left after administration but I still follow Rangers and they continue to mean a lot to me."