Brady completed 26 of 48 passes for just 216 yards and threw two touchdown passes, in addition to his turnovers, in the 27-24 loss on Monday at Raymond James Stadium.
Rams quarterback Jared Goff completed 39 of 51 passes for 376 yards, three scores and two interceptions in the victory.
"I don't want to ever take for granted how hard it is to win in this league, but we have high expectations," Rams coach Sean McVay told reporters. "I have a lot of confidence in our players."
The Buccaneers punted on the game's opening possession before Goff connected with Robert Woods to give the Rams a 7-0 lead 11 minutes into the game.
Tampa Bay responded with a 10-play scoring drive, capped off with a Brady touchdown toss to Mike Evans at the start of the second quarter. The Buccaneers also scored on their next drive, when Leonard Fournette ran for a two-yard touchdown to give Tampa Bay a 14-7 lead.
Goff then put together an eight-play, 70-yard touchdown drive. He threw a seven-yard score to rookie Van Jefferson to tie the game 4:37 before halftime. Matt Gay later made a 38-yard field goal to give the Rams a three-point edge at the break.
Neither team could do much on offense in the second half. Goff threw an interception on the first drive of the third quarter. Buccaneers kicker Ryan Succop then made a 38-yard field goal to tie the score at 17-17.
Jordan Fuller intercepted Brady on Tampa Bay's next drive. The Rams then put together a five-play, 42-yard touchdown drive. Rookie running back Cam Akers caught a four-yard touchdown toss from Goff to end that drive and give the Rams a 24-17 lead.
Los Angeles and Tampa Bay then exchanged punts for the next six drives before Goff threw a fourth-quarter interception to Jordan Whitehead.
The Buccaneers capitalized with a seven-play, 44-yard drive. Brady threw a 13-yard touchdown toss to Chris Godwin to end that possession. The Rams responded with an eight-play drive, which they capped off with a 40-yard Gay field goal to earn their final three-point advantage.
Fuller picked off Brady for a second time on the Buccaneers next drive, which allowed the Rams to run out the clock.
Rams wide receiver Cooper Kupp had 11 catches for 145 yards in the win. Robert Woods had 12 catches for 130 yards and a score for Los Angeles.
The Rams (7-3) host the San Francisco 49ers at 4:05 p.m. EST on Sunday at SoFi Stadium in Inglewood, Calif. The Buccaneers (7-4) battle the Kansas City Chiefs at 4:25 p.m. EST on Sunday at Raymond James Stadium.
"We've got to get ready to play a great football team coming up," Brady said. "Nothing has been great to this point.
"We've had some good wins [and] had some tough losses. Everything is about one week here in the NFL. We're going to have to go play really, really well against a great football team next week."