Luke Charman's first-half strike gave the home side a narrow lead at a wet and windy Whitley Park before Yannick Touré's late double sealed three important points for United's second string.
The result extends Newcastle's lead at the top of Group H to four points following their penultimate group clash with the Saints - and therefore saw United book their place in the knockout stages for a consecutive year after reaching the quarter-finals last term.
Dawson's side made it two cup wins in one week, thrashing Blyth Spartans 7-2 to reach the Northumberland Senior Cup semi-finals on Tuesday, but despite netting ten goals in two games, the coach revealed his satisfaction in keeping their Premier League 2 opponents - who occupy top spot in Division Two - at bay.
"We had a good result earlier in the week and scored a few goals but we were disappointed to concede," Dawson told nufc.co.uk. "Before the game, we set a challenge to keep a clean sheet knowing if we could do that, we would have a good chance of winning the game and progressing in the competition with a game to spare. I'm really pleased that we've done that and restricted our opponents to one or two opportunities, as well as scoring some good goals at the other end.
"We were disappointed that we didn't get a better result down at Notts County (a 2-2 draw), which put us on the back foot knowing we had Southampton and Leeds left to play, who have been two of our toughest opponents this season. Therefore, it was important that we got back on track, going back to producing a good performance which has ultimately led to a great result.
"It is one of our best wins this season. We have said how difficult Southampton have been as an opponent in the three games we've played against them. Keeping a clean sheet and scoring three goals against a team that don't concede many is right up there."
The conditions played a minor part in the cup clash, with strong winds and heavy rain in the second half making it difficult for both sides in their respective bids to pick up vital points.
However, Dawson was pleased with his side's performance for the full 90 minutes, adapting to different tactics than usual as United look to confirm top spot in their final group match against Leeds United at Guiseley's Nethermoor Park.
"The way Southampton play has been very effective," Dawson added. "They have been strong and defensively sound but I thought the change in shape and system helped us, with it almost going man for man.
"That provided our side more opportunities to get in behind the back four which we managed to create and take. When it's a cup competition and a lot is riding on it, getting the first goal meant they had to open up more. We were very disciplined and kept our shape, continuing to play some good football despite the conditions.
"As much as we had the wind on our side (in the second half), it was important that we just continued to play the same game. Teams can get caught thinking you have the wind behind you, pushing right on and then getting caught with a sucker punch at the other end so we stressed the importance of continuing to do what we did really well in the first half."
