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."