The Warriors trailed 17-7 at the interval, before producing a second-half turnaround to secure the result on the road.
McDowall admitted that Munster had made an impressive start, but was pleased with the way his side grew into the contest.
"Yeah, we knew we were playing into a bit of wind," McDowall told Premier Sports.
"We chose to play into the wind first half in the hope that it would stay strong for the second half, and thankfully it did.
"And yeah, we just talked about sticking to our process. For the first 20 minutes, Munster were unbelievable. They probably shocked us a little bit. A few of us haven't played yet this season.
"I thought that as the first half went on, we grew into the game and we started to hold on to the ball. We just talked about holding on to the ball, being direct, and it paid off at the end."
With a number of Warriors making their first competitive appearances of the new campaign, McDowall acknowledged that Glasgow's attacking play was not always as fluid as they would have liked.
However, the Scotland centre was encouraged by the way his team-mates responded as the match progressed.
"Yeah, we talked about it before in the changing room. It wasn't going to be perfect," he said.
"A bit of lack of cohesion maybe from boys who haven't played, but, you know, it's the first game of the season. It's always going to be like that.
"But I'm just really proud of the way the boys stuck to it, especially on someone like Dobie's hundred. He's a massive part of the club, and I think the boys showed how much he means to them today."
Jamie Dobie marked his 100th appearance for Glasgow Warriors with two tries, the Scotland international once again showcasing his versatility in the back-line.
McDowall was quick to highlight the consistency Dobie has brought throughout his Warriors career.
"Yeah, look, you always know what you're going to get from Dobie," he said.
"Luckily, it's a nine or ten out of ten every time!
"So, yeah, we're blessed to have players like that, that can play multiple positions. But yeah, just really happy for him. We hopefully made it a good week for him right the way through."
Having topped the regular-season standings in 2025/26 before falling to the Bulls by a single point in the semi-finals, Glasgow enter the new campaign aware of the challenge awaiting them each week.
For McDowall, the competitiveness of the league means there is little opportunity to look too far ahead.
"Yeah, I think so. Look, this league's so competitive," he said.
"I watched the games earlier today. There's no bad teams in this league; there never has been.
"So, especially coming away to a place like Thomond Park, you know you've got to be on your game.
"But, you know, we just said at the end there, we'll keep our feet on the ground and we'll rock up to training on Monday, work hard and go again next week."
The Warriors will now turn their attention to their first home match of the campaign, with Ulster the visitors to Scotstoun.
The fixture will also give Glasgow the opportunity to run out competitively on the new Scotstoun pitch for the first time.
"Yeah, massively looking forward to it," said McDowall.
"We did captain's run for the first time on Friday and it was lovely.
"So we're looking forward to running out and hopefully giving it a good first run."