Eldin Jakupovic's botched clearance gave Wanyama the chance to try his luck from 40 yards after just three minutes at the KC Stadium and the Kenyan midfielder produced a wonderful finish to secure a 1-0 win.
Saints have now won 10 of their last 11 matches in all competitions and lie second behind Chelsea.
The buoyant away fans were chanting about a title challenge at the final whistle but while Koeman sees that as a fanciful notion, he does not feel Europe is beyond his side.
And on this form, the Champions League may be a more realistic aim than the Europa League.
Asked if he could deliver the title push the supporters half-jokingly sang about, the Dutchman said: "That's always positive but that's not what will happen.
"But to continue winning and being up in the table, why not?
"Expectations are high but I like that because we would like to win something.
"Football players like to win games and if you win games, you will have good possibilities.
"Maybe to play in Europe, you never know
Everybody knows we have to work hard every day and they players are incredible in the way they are acting together.
"If we keep like this, it can be a fantastic season."
Koeman also had a special word of praise for his match-winner.
"(Wanyama was) fantastic
The goal was amazing but it was how he played everything in the midfield.
"He was strong, he won the battles and played good on the ball.
"It's fantastic to have that kind of player."
Southampton's success has been built around a solid defence, with six clean sheets in 10 league games and a miserly five goals conceded.
"We worked hard to have good organisation in the team because clean sheets bring points," stressed Koeman.
"It's important to have that and we've received only five goals until now
That's one of the keys to be successful."
Mauricio Pochettino's Southampton were Hull's bogey side last season
doing the double over the Tigers, and boss Steve Bruce was no less impressed by Koeman's much-changed squad.
"They were going to be everbody's whipping boys this season apparently, but they've bought well and the manager needs a pat on the back," he said.
"Arguably they're as good as they were last year
It looks as if what they've let go has been replaced with real quality.
"Southampton are a very good side and it's no fluke that they're at the top end of the division."
Bruce admitted the early error by Jakupovic, his third choice keeper making only his third Premier League start, was a decisive moment.
"It's disappointing to lose a game by a mistake again," he added
"It's a great finish by Wanyama but you make a mistake like that after a couple of minutes and it sets everybody back on their heels.
"To go 1-0 down to a goal like that is difficult to take for everybody, it shocked the whole stadium, the players and everyone."
Source : PA
Source: PA