After a chaotic week in which Andre Villas-Boas was shown the door, Spurs secured a 3-2 victory at St Mary's courtesy of Emmanuel Adebayor's brace and a Jos Hooiveld own goal.
It was the first win of Sherwood's temporary reign after Spurs exited the Capital One Cup against West Ham in his only other match in temporary charge.
The result and performance at St Mary's have seen odds on him being handed the hotseat permanently shorten, with the former midfielder planning to talk to chairman Daniel Levy about the situation sometime on Monday.
Glenn Hoddle, Louis van Gaal and Frank de Boer have also been linked with the vacancy, for which Dembele was keen not to comment when asked whether the players wanted Sherwood to get the post full-time.
"I don't think I am going to talk about this now, but we have a good team," the Belgium international said.
"If everybody is motivated we can do big things, and you saw that in the second half.
"I think we are a very good team and it is important we win the game for anyone.
"I think with our strength we have to do big things, so that is why we are happy we won."
The three points means Tottenham head into Christmas seventh in the Barclays Premier League and halted a run of back-to-back defeats, having lost to West Ham and been humiliated 5-0 by Liverpool.
"It was very important for us," Dembele said. "I think there was a while that we won a few games, like three games in a row I think, then we had Liverpool.
"After that we played good against West Ham as well, even though we lost it.
"Against Southampton I thought the second half we played well, nice attacking football and were organising so it was very good.
"It was very important. Everybody knows to come to Southampton and win the game is very difficult. They are a very good team."
Dembele was forced to watch the second half at St Mary's from the sidelines after twisting his ankle.
Despite the knock, the 26-year-old expects to be available for the Boxing Day clash with West Brom - a match in which Adebayor could well again prove key.
Cast aside by Villas-Bas, the Togo striker has started both matches under Sherwood and repaid that faith with three goals.
Asked if Adebayor's rejuvenation was like a new signing, Dembele said: "Yeah, it looks like that. I think against Southampton, and the last game as well, he was very important for us.
"He scored his goals but as well as that he played a good game.
"You can play the ball to him and he will keep it, he is going to do something with it. That is very important for any team."
Adebayor proved a nuisance throughout the afternoon for Saints, who have now gone six matches without a win.
Tough-looking trips to Cardiff and Everton now await, but James Ward-Prowse is confident they will soon come good.
"We're disappointed to lose again," the 19-year-old midfielder said. "I think we're all missing the winning feeling.
"I think we were just beaten by a good Tottenham Hotspur side, who were obviously galvanised under the new manager.
"We knew before the game it would be difficult. Obviously, they're trying to prove a point to the new manager and I think they did that.
"These sorts of things happen during the season, but it's a test of character for us, and I know among us we're mentally strong enough to get through that."