The 27-year-old, who has scored 40 times in 332 appearances across the board for the Lilywhites since signing from Cork City in 2014, opened the scoring in the 20th minute at the Aviva Stadium in Dublin.
Tottenham Hotspur forward Troy Parrott added a second for the hosts eight minutes later before Swansea City's Michael Obafemi made it 3-0 early in the second half to help The Boys in Green break their duck at the third attempt in Group One.
After forcing the ball past Craig Gordon from close range — once Shane Duffy had headed James McClean's corner back across goal — the midfielder said: "The gaffer [Stephen Kenny] just told me to get myself in and around the box. I was sniffing about and thankfully the ball came to me and it found its way into the back of the net.
"I didn't know too much about it. That was the all important first goal, though."
Ireland had lost both of their opening fixtures heading into their clash against the Scots. They were beaten without scoring, by an identical scoreline, away in Armenia and at home to the Ukraine.
Browne, who was expected to earn his 23rd cap for his country in Tuesday evening's game against Oleksandr Petrakov's men at the Stadion Miejski LKS in Lodz, Poland, said: "Tonight was brilliant. Given the start we've had to this campaign, it was a big chance for us to kick-start what we've wanted to do from the beginning.
"Thankfully we got the right result tonight, with the performance to match that.
"At this level you've got to be motivated for every game. At the start of this campaign we've been on the wrong end of two results. It was such a shocking start.
"Tonight was really important for us to get some points on the board. The result is all that matters, thankfully we got the three points."