Saints moved six points clear of the relegation places after securing their first win since Pochettino took charge at St Mary's.
City boss Roberto Mancini described his side's performance as "worse than poor" but Davis says the Saints manager's tactics worked perfectly against their title chasing opponents.
"We're obviously delighted with the result," said Davis. "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."