The Saints saw off the defending Premier League champions 3-1 at St Mary's stadium to mark Pochettino's first victory as manager since he replaced the sacked Nigel Adkins last month.
The Argentinian says teams can only win if they have belief and feels it was abundant in his hard-working Southampton side on Saturday.
He told the club's official website: "Victories only come if you believe you can get them. Everyone here at Southampton has belief.
"The key is all of the work we put into it and the work that we've put in for all of the matches.
"Today we've seen it again, and the result has come to us too.
"I have said before in the other matches against Everton, Manchester United and Wigan that we played well but didn't get the points that we deserved. Today we played well and got those points that we deserved."
Argentinian Pochettino's arrival at St Mary's was met with hostility from some Southampton fans following Adkins' shock dismissal but the boss was full of praise for the home support after the win.
He added: "We're all very happy. I want to thank the supporters for the great support they gave us today.
"I have said many times before that I was very well received here, I have had a great welcome and I am very happy with the supporters, all the staff and the players that are here."