And while Ulster played well, Saints were left to rue their own errors as they slipped to a third defeat in as many Champions Cup matches this season.
"We weren't a million miles away at the end but we just gave away too many points," said Dowson, who took charge of Saints with Chris Boyd sitting in the Barwell Stand as he served the first of a two-game ban.
"We defend for long periods of time, and a lot of the time we do it well, but eventually some of that pressure tells and when we get into those attacking spaces we don't make as much of it.
"We need to be more consistent in turning pressure into points.
"We have to start learning those lessons or we're still going to fall away on a four-point loss.
"We need to improve - and we need to improve quickly week to week.
"We need to make sure we're better in scenarios where we can exert pressure.
"It's about training it, understanding what you need to do and then training it under pressure again and again until you can't get it wrong.
"It's very hard to recreate scenarios midweek because the injury risk and fatigue mean you don't want to create that pressure physically, but you do have to try to create it and offer those opportunities.
"But the best and hardest way to learn is in games like today.
"Ulster are a physical, direct team and their forwards go route one and our boys stood up.
"A lot of the time we put good defensive sets in and didn't give away too many dumb penalties, but we've come out on the wrong side so we need to be better."
Courtney Lawes led Saints out on his 250th appearance for the club and went on to produce another huge performance.
"He's an extraordinary player and an extraordinary person," Dowson said.
"He's more often than not outstanding, and to be that good for that long is impressive.
"We talked about it before the game because we wanted to make sure we showed him how special he is.
"Alex Waller said before the game that he's one of the best players in the world and that's not just the physical element, it's also the psychological element to be that good for that long."