The winger struck on the hour mark to notch his fourth of the season having seen a header chalked off in the first half.
A number of other chances came the hosts’ way, but Puncheon was just happy to take three points from a side breathing down his team’s necks.
"We were comfortably the best team and we could have won it by three or four, but we didn’t really take our chances," he said after the game.
"The point was to go out and get three points, and that’s what we’ve done today.
"It’s good to get back onto the run we were on for the four games before Liverpool (where Saints lost 1-0 last weekend)."
Puncheon is now Saints’ second-highest goalscorer this season and has made more appearances than he did in the whole of the 2011-12 campaign.
And the winger says he is glad to finally be making a big impression at St Mary’s almost three years after he joined the Club.
"It hasn’t taken some time, but I wouldn’t change the way things have gone because it’s worked out well," he insisted.
"The Club is in the Premier League where they said they were going to be and I’m playing, so that’s the main thing.
"I’ve matured as a person and as an adult, and those things are helping on and off the pitch – and that’s showing in my performances."
Saints now have a two-week break between fixtures, with next weekend’s trip to Chelsea having been postponed because of the Blues’ continental commitments.
And, with team captain Adam Lallana suffering a knee injury during today’s game, Puncheon says the extra recovery time will serve the squad well.
"It’s always good to have a break when you’re in the Premier League because it’s the most exhausting league, both mentally and physically," said the 26 year-old.
"That break will do us well going into the Sunderland game at home. We’ve got a good squad of players, and that’s going to be needed over the Christmas period.