Adams collected the ball on the edge of the area and turned before chipping the ball into the top corner and out of reach of the goalkeeper in the 19th minute.
Southampton had 12 shots but just one was on target as they secured a first away Premier League win since March.
“We are absolutely happy with the win, I think it was a very deserved win,” Hasenhuttl said.
“We had a very good performance in the first half where we had only one criticism – that we didn’t score more often. The game could have been finished at half-time with three or four to zero.
“The only thing I have to criticise in the second half is that it’s normal when you’re only 1-0 up to settle on the back foot and we have been a little bit too deep in these moments.
“Finally we made the three points and it’s an important win for us, absolutely.
“We’ve definitely had games where we could have scored more often, but the good thing is when you have more clean sheets then it’s easier because then sometimes one goal is enough.
“When you have to score two or three goals to win a game because you know you always concede one or two, this is even harder I think.”
It was a less eventful game for Claudio Ranieri after his Watford side shipped five goals against Liverpool only to score five themselves in his second game against Everton last weekend.
The Hornets remain without a home win since the opening day of the season, and despite their emphatic 5-2 win at Goodison Park are slipping closer towards the bottom three, with the gap now just three points.
Ranieri urged fans to continue their support and wants Watford to find some consistency.
“We had to improve our defensive way together, I know,” the Italian said.
“I’ve been working here three weeks and the Southampton manager is working since 2008 and that will make a difference, but I think we were too passive in the first half.
“The second half was much, much better. They put them under pressure. Maybe we deserved to draw the match because we had the chance with Ismaila Sarr and at the end, but that’s it.
“But we must be patient and our fans must be patient and take us with love, support us at this first moment because it is very, very important.
“I don’t want us to be up and down in the table…I would like us to be more consistent.”
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