"It's sweet," Belichick told CBS. "Everybody counted us out from the beginning of the season, mid-season, but we're still here."
Belichick and Brady first combined for the Vince Lombardi Trophy in 2001. Wins followed in 2003, 2004, 2014, 2016 and Sunday night's at the Mercedes-Benz Stadium in Atlanta was the latest success.
The lowest-scoring contest in the 53rd edition of the NFL's showpiece final was built on defence - and Brady's experienced leadership.
The 41-year-old threw for 262 yards and engineered the game's only touchdown drive as Sony Michel punched the ball in to give New England the lead midway through the fourth quarter.
Brady told CBS: "It was an unbelievable year. We just fought through it more so than anything.
"It's unbelievable to win this game. They played so well, the Rams' defence they played their butts off.
"What a great defence, they had a great plan. They made it tough on every play.
"We just kept fighting and finally got a touchdown. The (Patriots) defence played the best game of the year."
Julian Edelman - voted MVP with 141 yards receiving - and Rob Gronkowski also shined for New England.
The pair had 228 of New England's 262 receiving yards, with both nabbing key receptions in the touchdown drive.
Starting at their own 31, the veteran quarterback found Gronkowski with a short pass for 19 yards, then, out of the gun, hit Edelman for a 13-yard completion.
Now at the Rams' 38-yard line, Brady found Rex Burkhead with a short seven-yard pass before finding 87 again as Gronkowski beat two defenders to finish two yards shy.
This was the only redzone visit of the night and Michel took advantage of good blocking to score his sixth post-season touchdown.
Plaudits should also go to the New England defence, who contained Jared Goff to just 229 yards and the Rams to just three points - having scored an average of 32.9 per game in the regular season.
The strong Patriots defensive performance was capped off when Stephon Gilmore read Goff and secured a pick on the New England two-yard line with just over four minutes remaining.
Greg Zuerlein's field goal was the Rams' only score all night as the Patriots found the end zone for a 10-3 lead with seven minutes left.
Stephen Gostkowski made it a two-score game late in the fourth when he converted a 41-yard attempt.
For the Rams, it was do or die - and there was no way through the Patriots.
Rams head coach Sean McVay took responsibility for the defeat.
"I just never enabled us to get into a rhythm offensively," he said on nfl.com.
"They did a good job, and it seemed that any time we got a little bit of a positive play then we'd end up having a penalty or move ourselves back.
"So, I think a lot of it is a result of some of the things they did but then also the play selection.
"I was not pleased at all with my feel for the flow of the game and kind of making some adjustments as the game unfolded and with giving ourselves a chance at some success and put some points on the board.
"Credit to them, they did a good job, and I certainly didn't do enough for us."
Goff added: "I'm mad at myself. I wish I would've done things differently. I wish I could've made extra plays.
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