The winger scored four minutes after coming off the bench at the Stadium of Light earlier today, cancelling out Phil Bardsley's 68th-minute opener for the hosts.
Saints are now practically assured safety by virtue of the fact that their vastly-superior goal difference should keep Puncheon and his teammates out of the drop zone regardless of the results on the final day of the season.
Mauricio Pochettino's side has stuttered a little in recent weeks, with today's draw stretching their winless run to five games.
However, Puncheon is confident that today's result has seen the team over the line and will send them into next Sunday's game against Stoke in a positive mood.
"It's a nice moment for the whole football club, for the fans and for everybody who's been involved in this," he reflected.
"It's been a bit edgy towards the end, so maybe people have panicked a bit but we have done it now unless something dramatic happens. We can be happy and go into the Stoke game at home to get three points from that."
Saints had dominated proceedings in the north east prior to Bardsley's goal, with Sunderland goalkeeper Simon Mignolet in particularly good form.
Puncheon has been limited to substitute appearances in recent weeks, and today he made the ideal impact after only four minutes on the pitch.
The winger got his head onto a cross from fellow replacement James Ward-Prowse and, after Mignolet had parried, managed to poke in his seventh goal of the season to level matters.
"When you're not playing, you obviously want to do the best you can when you come on," Puncheon said after the game. "I tried to do that, and we've ended up getting a point.
"I think that is the first goal I've scored from the bench.
"As soon I headed it, Rickie [Lambert] shouted 'Punch!' so I've naturally switched on to the rebound and got there. That was the most important thing.
"Overall, we had chances in the game but sometimes in football Lady Luck doesn't go for you," he added.
"You just have to keep plugging away, and that's what we've been doing. In the last few weeks – when we haven't looked like scoring – we've just kept going."