Gay drilled a 30-yard field goal as time ran out to give the Rams a thrilling 30-27 win after Brady and the Bucs had recovered from 27-3 down to tie the game at 27-27 just moments earlier.
The win means the Rams will advance to an NFC Championship game at home to the San Francisco 49ers next week where a place in the Super Bowl will be at stake.
The loss meanwhile ended the 44-year-old Brady’s hopes of adding an eighth Super Bowl title to his record haul of seven Vince Lombardi Trophies.
“We’re a resilient team,” Rams quarterback Matthew Stafford said. “We knew it wasn’t going to be easy.
“We sure let them back into the game with a bunch of mistakes on offense. Our defense played outstanding.
“But we’re tough a team, this is what we’re all about. Just happy to get a win and keep moving,” added Stafford, who finished with 366 passing yards from 28 of 38 attempts.
Brady, who was sacked three times and hit on six other occasions by the Rams defense, threw for 329 yards with one touchdown.
But the Rams very nearly were left facing up to the end of their season after an error-strewn second-half display allowed Brady to mount a comeback for the ages.
The Rams had dominated the Buccaneers through the first half and early part of the third quarter to take a 27-3 lead.
Stafford threw for two touchdowns to Kendall Blanton and Cooper Kupp before rushing for a score of his own to give the Rams a 24-point lead midway through the third quarter.
But just when it looked as if the Rams were home and dry, errors crept into the visitors’ game and enabled Tampa Bay to come roaring back.
A Kupp fumble led to a Leonard Fournette touchdown to make it 27-13.
Gay then missed a field goal that would have extended the Rams lead, and Brady soon afterwards picked out Mike Evans with a 55-yard touchdown reception to make it 27-20.
The sense of a Rams implosion deepened when running back Cam Akers then fumbled on the Los Angeles 23-yard line.
That gave Tampa Bay the ball back in prime territory, and seven plays later Fournette rumbled over from nine yards to set up Succop’s game-tying extra-point.
The momentum appeared to be all with Tampa Bay, but the Rams responded superbly, with Stafford making a 20-yard pass to Kupp before finding his favourite receiver again with a 44-yard completion.
That left Gay a chip to win the game from the Tampa Bay 12-yard line.
Stafford praised the contribution of star receiver Kupp for the two late catches that allowed the Rams to snatch victory.
“I’ve spent a lot of time with the guy and know what he’s all about,” Stafford said. “He’s the heart and soul of this offense. He’s an unbelievable competitor.”
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