The Warriors had not returned to TD Garden since winning Game 6 of last season's NBA Finals to secure the championship.
Much like in that series, the Celtics were littered with turnovers (17 in total resulted in 24 points for the visitors) and struggled to fire on all cylinders offensively.
This time, though, Boston found a way to win. Trailing by 11 points in the second half, the Celtics fought back and Brown tied the game via a three-pointer with 18 seconds remaining and then, having gained an eight-point advantage, held on after a sloppy ending to overtime.
"That felt like a playoff game," Brown said.
"Their intensity, their force where they came, that's a game I'm sure they wanted to win. We were down, what, five to seven [points] in the fourth quarter, four, five minutes left.
"To be able to have poise to battle back, that shows a lot of growth. We're taking steps in the right direction."
Jayson Tatum had a career-high 19 rebounds to go with 34 points and six assists but was also guilty of a couple of poor turnovers.
He too, though, felt the Celtics showed the sort of requisite desire to win when the team is not particularly playing their best basketball.
"You're going to need games like this," Tatum said.
"There's going to be a handful of playoff games where you don't necessarily shoot the ball well, maybe even on the road – under 40 per cent, 73 per cent from the free-throw line, 17 turnovers – and still find a way to win.
"I think that just shows the depth of our team, that on a below-average night for us, we can still find a way to win. That's all that matters at the end of the night. Did you win or did you lose?"
Interim Celtics coach Joe Mazzulla felt the nature of the victory showed how the team have grown since losing the Finals.
"For us, we've talked about poise and physicality," he said.
"Everybody asks, what did we learn [from losing in the Finals]? What have we learned? And I think what we've learned is it takes a mindset in order to be successful. And it takes a mindset to be a really, really good team in the NBA. And you can't be inconsistent with that.
"I think even though you play really, really well and we win, we're still going to have some of those moments. Yeah, we had some turnovers, we had some kind of plays where it's just kind of like, 'That shouldn't happen'. But that's going to happen.
"It's about just the habits that we're growing as far as our mindset, our poise, our physicality, our ability to execute, our ability to handle the chaos of an NBA game."
The Celtics sit top of the Eastern Conference with a league-best 34-12 record, while the Warriors are ninth and 22-23 in the West.