Saints played second fiddle us in a one-sided first 45 minutes but, crucially, as Glenn Hoddle later pointed out, we failed to cash in our dominance.
James Beattie's header in the second half was enough to decide the match as Saints made it a miserable return to the south coast for Glenn, John Gorman and Dean Richards.
Boss Strachan said: "We gave ourselves a lot of problems, we were playing scared football hoping they would make mistakes rather than trying to win the game.
"After half-time the game became a bit easier for us, there was a big difference and the players became mentally brave and physically brave.
"Their goalkeeper made world-class saves, absolutely phenomenal saves, but then we scored by one of our men being extremely brave and he deserves that for all the hard work that he has put in since I came to this club."
Gordon Strachan said Southampton had been too scared to play football against us in the first half of the Boxing Day clash.