The black, green and gold were level at 6-6 at half-time and then moved into the lead as Tom Pearson's superb assist set up Curtis Langdon for a strong finish.
Newcastle kept fighting and could have levelled things up with a conversion following Iwan Stephens' score in the corner.
But Brett Connon narrowly missed the tricky attempt from the tee, and Saints eventually held out with Tom James winning a breakdown penalty to earn his side their first Gallagher Premiership win of the season.
"I'm delighted with lots of aspects of it in terms of our defence, sticking with it at the end and not giving away a penalty," Dowson said.
"The last few minutes weren't relaxing in any way!
"I don't think we were too stressed (about getting the first league win). We had the opportunity to win at Sale but didn't quite get over the line in the last minute, and the same against Bristol.
"But the feeling against Bristol was that we hadn't been ourselves for large parts of that game and that was the frustrating thing.
"I don't think there was any stress around it, but there was a concern that we had to get more of our game on the pitch for longer periods.
"If we'd had the right field position in the first half, we would have tried to put a bit more together, but clearly it was raining a lot heavier in the first half than the second half which makes it difficult and creates a game like that.
"We spoke about our discipline at half-time, penalties giving them territory and that meaning you can't play so some of those discipline issues were meaning we were playing a lot of the game in our 22 and being put under pressure by Falcons.
"There weren't going to be many opportunities today to convert pressure and we hadn't been very good at Sale at converting what we did have.
"It's clearly been a focus for us and the few opportunities we did have here, we either kicked points or got that one score through Curtis."
Pearson was the one who opened the door for Langdon, making a superb break and offloading brilliantly out of the back door for the onrushing Saints hooker.
And Dowson was full of praise for Pearson, who moved to Northampton following the sad demise of London Irish during the summer.
"You could see last year the way he plays and the ability he has - and I think there's loads more to come," Dowson said.
"He was in the England squad with Borthers (Steve Borthwick) and they were very positive about what he can bring to the table.
"Now it's just about getting him playing regularly, enjoying his rugby again, getting set up with the Saints and all those guys who came out of the Irish squad - Chunya Munga and Tarek Haffar - are very talented and very ambitious as well."