Brady completed 23-of-35 passes for 304 yards as the Patriots consolidated top position in the AFC East and moved to a 5-3 record before 84,004 fans at Wembley Stadium.
Things started brightly for the Rams when on the opening drive quarterback Sam Bradford hit rookie receiver Chris Givens on a 50-yard scoring pass.
But the Patriots hit straight back on an eight-play 78-yard drive which culminated with Brady hitting Brandon Lloyd in the corner of the end zone from 19 yards.
After the Rams were forced to punt, Brady hit tight end Rob Gronkowski on two 25-yard completions before running back Shane Vereen went in for the go-ahead score on a fourth and one from the one-yard line.
It became 21-7 to the Patriots midway through the second quarter when Brady hit Gronkowski on an eight-yard scoring play and the host team Rams suffered a further blow when a fake field goal from the Pats 35 was stymied inside the two-minute warning.
That allowed Brady enough time on the clock to march his team down the field, and a pass interference call on Bradley Fletcher in the end zone placed the ball on the one and allowed Stevan Ridley to plough over to give his side a 21-point half-time advantage.
The procession continued after the break with Brady immediately masterminding a six-play 80-yard drive that ended in a second TD hook-up with Lloyd, this time from nine yards, and a 35-7 lead.
Stephen Gostkowski added a 26 yard field goal before New England added an exclamation mark to the scoreline with 11:23 remaining as Brady found Gronkowski for another six points on a 14-yard pass.
The Rams dropped to 3-5 at the halfway point of the season and are at the bottom of the NFC West.
The NFL has scheduled two regular-season games to be played in London in 2013 - the San Francisco 49ers versus the Jacksonville Jaguars and the Pittsburgh Steelers against the Minnesota Vikings.
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