Puncheon turned in a fine display against Rangers, with whom he had an unsuccessful loan spell last season, and netted the second goal for the visitors.
"It's always good to get one against a former club, but the most important thing was getting three points," he said.
"We sort of got on the front foot and did what we've been doing in recent weeks. I thought we were solid, and that showed in the first half and the second half.
"We have always known as a group that we shouldn't be where we are.
"There have been games that haven't gone our way and games that have slipped through our hands, but we were determined not to do that from back-to-front and in [terms of] finishing off goals and that happened.
"You can sit there and say it's easy, but we were on the top of our game on every yard of the pitch.
"The two midfielders covered every blade of grass on the pitch, Gaston [Ramirez] and Rickie [Lambert] both put in a lot of work as did the full-backs, centre-halves and goalkeeper.
"That counts, that's what it's all about and that's what we did."
The result helps alleviate the pressure mounting on manager Nigel Adkins, whose future has come under increasing scrutiny in recent weeks.
Southampton have found themselves in the headlines for the wrong reasons as a result and Puncheon admits it has been hard to ignore.
"It's natural, we're human beings at the end of the day," he said.
"Just because we're professionals it doesn't mean that we don't see and hear things. We've got eyes and ears.
"That's part-and-parcel of football so you've got to take that on the chin and turn up on a Saturday."