Morgan Schneiderlin's early strike had given the hosts a half-time lead before second-half goals from Rickie Lambert and Jay Rodriguez had the Saints seemingly cruising to three points.
Norwich threatened a late comeback as quick-fire goals from Johan Elmander and Robert Snodgrass gave them hope of snatching a point, but a swift break through Steven Davis culminated in Sam Gallagher slotting home his first Premier League goal in stoppage-time.
Although the visitors had rarely threatened prior to their two goals with Southampton looking comfortable, Pochettino admitted his relief at being able to hold off the late scare.
"This is part of football," he told Sky Sports. "Basically, we were dominating throughout the entire game - we were 3-0 up.
"In football, especially in the Premier League when you let your guard down, these things can happen. We conceded two soft goals at the end.
"That can give you a very nasty surprise at the end but overall I felt we dominated and we fully deserved the win in the end.
"He who has played in football knows that these things can happen but, overall, the most important thing today is that there wasn't any further damage and we got the three points."
Pochettino was also pleased to secure the win in front of a home crowd having only previously managed to do so once from their last eight Premier League outings at St Mary's.
"I think that it was really important that we got the victory today at St Mary's," he said.
"It was really important to get back to winning ways here at home. Overall, it was fully deserved.
"It was quite a vibrant game, it was a very good game and we fully deserved the three points."
Asked why Lambert started on the bench, Pochettino replied: "Because I have to make decisions as a manager, that's why. I'm happy with the performance of all the players."
Roy Hodgson was in attendance to monitor the performance of England hopefuls Lambert, Rodriguez, Luke Shaw, Adam Lallana and John Ruddy.
Southampton's English contingent did impress on the whole, but Pochettino says that maintaining the team's good run of form will give them the best chance possible of making Hodgson's World Cup selection.
"It's vital that all players are focused, that we keep on training well and we keep on winning games," he said.
"Only by doing those things, the players who are possibly in the English national side will have more chance of being on that plane to Brazil.
"If we do all of those things, they will have a greater chance of going to the World Cup."