Koeman's side were unfortunate to go down 2-1 to United in the reverse fixture at St Mary's Stadium last month, as Robin van Persie scored with each of the visitors' two shots on target.
That setback for Southampton came in the middle of a five-match losing run in all competitions, which saw the club slip down the Premier League table and exit the League Cup courtesy of a quarter-final defeat to League One side Sheffield United.
However, the south-coast outfit have since got back on track, with an impressive 2-0 victory over Arsenal on New Year's Day cementing their place in the top four.
While acknowledging the improvement United have shown in an 11-match unbeaten streak, Koeman believes Southampton can claim a notable victory this weekend and leapfrog their third-placed opponents in the table.
Speaking at a news conference on Friday, Koeman said: "I think the victory against Arsenal was important and important was the one point as well against Chelsea [on December 28], because we didn't have that kind of result [against a big team] before that game.
"It’s a quality of them [United] that they don't need a lot of shots on target to score two goals and to win games. And if you analyse them in the last few weeks, you can see that they [have] grown up, the players have more confidence in the system, they have less injuries than they had before.
"It will be a tough game but it’s a nice one because the pressure is on them and we have to show that we believe in a good result. I am full of confidence."
Jack Cork is available again for Southampton following an ankle injury, while Koeman also expects England right-back Nathaniel Clyne (also ankle) to be fit.
Koeman could hand a debut to on-loan winger Eljero Elia, having revealed: "He will be part of the squad for this Sunday, for sure."
However, Sadio Mane will miss out as a result of the calf problem that threatens to prevent him from representing Senegal at the Africa Cup of Nations.
United have seen their own fitness problems clear up in recent weeks and manager Louis van Gaal confirmed on Friday that Ashley Young (hamstring) is now the club's only sidelined player.
Rafael may not be risked after undergoing surgery on the cheekbone he injured in Sunday's FA Cup win at Yeovil Town, but Luke Shaw - another player withdrawn during that game - is available to face his former club and both Daley Blind and Marouane Fellaini are back in training.
United are set to have new signing Victor Valdes on the bench, the experienced former Barcelona goalkeeper having arrived at Old Trafford as back-up to David de Gea.
Van Gaal's men were held to a 1-1 draw at Stoke City in their last Premier League outing, but Chelsea's defeat at Tottenham later on New Year's Day ensured United still made up ground on the leaders.
They are now nine points behind Chelsea and Manchester City.
