The south-coast club led twice in the match through goals from Rickie Lambert and Morgan Schneiderlin but two headers from the Dutchman in the final minutes of the game ensured that it was United who took all three points.
And, although Adkins was proud of the way his team played against the 20-time English champions, he could not hide his frustration at seeing his side throw away a first win of the campaign.
"I've just seen the two goals and at the end of the day they're two free headers - you can talk about football all day long but you've got to defend better than that," the Saints boss told Sky Sports after the game.
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A similar situation occurred in Southampton's opening game of the season, when they led at Manchester City only to see the hosts score twice to come from behind and win, and Adkins was disappointed that his players were unable to learn their lessons from that defeat.
"It's alright talking about it but it's then applying it," he stated.
"We've changed the front three to try to give us more energy, the lads put a great shift in today and we're looking for the impact to come off the bench to give us that physical prowess you need but we're playing against top-quality players and, if you don't press the ball and close down sufficiently, they're going to pick a pass."
Despite not picking up any points from their opening three Premier League matches, the Saints boss is remaining calm and is positive that his side will soon get off the mark.
"We're going to remain positive, we knew looking at the first four fixtures that they were going to be challenging," the former physio asserted.
"[There are] loads of positives. The chairman's done great to bring a few more players in the transfer window, the lads will go on international duty now, then we'll come back, regroup and look forward to the next game.
"We want to play football a certain way, we want to pass it about. Rickie Lambert did well with the physical side of the game today, got a great goal which is something we've worked on in training, and I thought Morgan Schneiderlin's goal was an excellent one. James Ward-Prowse is an outstanding talent.
"We'll pick ourselves up again, as we always do.
"We're without a point, we're bottom of the league - that makes us the strongest team in the league as we're holding everybody up!" he joked.
"We've played Manchester City and Manchester United, we've managed to score goals, we just need to keep them out at the other end."