Dusan Tadic's second-half strike was enough to take maximum points and send Southampton third in the Premier League as the visitors won at Old Trafford in the league for the first time since 1988.
Southampton had to withstand plenty of United possession, but struck at the key time with the only shot on target in the game.
Koeman's side have taken seven points from meetings with Chelsea, Arsenal and United in recent weeks and the Dutchman feels his side fear nobody.
"We defended well and the spirit was unbelievable from the players, [we were] a little bit lucky but remember the [reverse] game, Man United was the lucky one," he said, referring to United's 2-1 win at St Mary's Stadium in December.
"Maybe we are [lucky] but if you finally win you deserve to win because I think it's a compliment to the players how they play.
"I said before we could make it very difficult for them and we did. We know we have a strong team and that was important at the end of December.
"We got a point off Chelsea, the three points against Arsenal, it makes for good belief among the players. That's always what you need if you are not a big club and you have to play a big one.
"We are not afraid of the name, or the stadium. I think we learned that in the last few weeks."
Koeman, however, was slightly unhappy about Tadic's celebration, with the Serb earning a yellow card for removing his shirt.
"I don't like that because it's a yellow card and a stupid mistake is then a red," he added.
"If it's in the last minute of the season and you're qualified for European football, then it's no problem, but I don't like that.
"Keep that shirt on. I can't change it on the bench but I can tell him now that I don't like that and I will do."
