Kieffer Moore opened the scoring after just 36 seconds at the Stadion Kraj Bistrice, with Wales netting inside a minute for the first time since Robert Earnshaw in 2004.
But the Dragons doubled their advantage with two minutes and 27 seconds on the clock, this time Harry Wilson found the back of the net with a fine finish from distance.
It was the first time Wales had scored twice within the opening three minutes of a game since April 1961 against Northern Ireland to give Bellamy a dream start.
However, the conditions in Montenegro were far from ideal, with the rain-soaked pitch allowing Driton Camaj to pull one back for the hosts in the 73rd minute.
Wales were able to see out the remaining minutes to seal their first win of their Nations League campaign, building on their impressive showing against Turkiye.
"These are the toughest conditions I've been in. Nothing about coming here has been easy," Bellamy said.
"You could have had a lot of excuses about this trip. The change of venue, the travel here, a bus driver who took his time, the conditions. But sometimes you learn from it.
"There is a reason why this group have qualified for major tournaments. It wasn't a football match. The conditions weren't going to allow it."
Wales are now unbeaten in their last eight competitive fixtures (W4 D4), their longest such run since a run of 11 between September 2019 and November 2020.
Bellamy's appointment has seemingly revitalised Welsh football, with his expansive and exciting style of play catching the eyes of those across Europe.
The Dragons return to action next month when they face Iceland and Montenegro once again, something that Bellamy is already counting down the days for.
"Last night I was really excited about the game but also sad as it ends tomorrow," Bellamy said.
"What happens on Tuesday? They're all going to leave me and I don't want them to go.
"You want your first win of course and you want it quick. Longer-term I see us being able to affect games in a positive way."
Harry Wilson's 36th-minute penalty earned Wales all three points in Monday's Nations League Group B4 clash at Cardiff City Stadium.
Bellamy, who succeeded Rob Page in July, is the first Wales head coach to avoid defeat in his first four games in charge of the senior men's side.
As well as beating Montenegro home and away, Wales have also drawn 0-0 at home to Turkiye and 2-2 in Iceland.
Reflecting on his side's latest win, Bellamy told S4C: "I was impressed by us being able to control the tempo of the second half, but we still need improvement on it.
"You will get breaks and it feels like you can score because they have players forward, but it's also dangerous then for the counter.
"That transitional game becomes a little complicated, whereas in the first half they sat a lot deeper which allows us to really control the tempo of the game. Of course at the moment that really suits us.
"But we have got to stay calm, in transition but even when we have knockdowns and balls in and around.
"You secure the ball and pass to your team-mates and there are still bits for improvement we need, but we were better than last time."
Match-winner Wilson has now been directly involved in nine goals in his past 11 appearances for Wales, as many as in his previous 41 matches.
The Fulham forward kept his place in a much-changed Wales side on the back of Friday's draw in Iceland, which showed seven alterations in all.
Wilson was the star performer in a game Wales dominated for large parts, with Montenegro failing to register a single shot on target.
It means the Dragons have avoided facing an effort on target in two of their past seven matches across all competitions, matching the tally from their previous 34 games.
For all Wales' dominance, though, Montenegro substitute Andrija Radulovic fired a 20-yard drive against the crossbar.
"There were a lot of changes, and a lot of players who all definitely deserved the opportunity to play as well," Bellamy added. "The Nations League is so good and so competitive."
Wales trail leaders Turkiye by two points heading into next month's contest in Kayseri, before the Dragons round off their Nations League campaign at home to Iceland.
Montenegro, bottom of the table without a point and on a record run of six losses in a row, host Iceland and Turkiye in their final two matches.
