The Patriots acquired a sixth-round pick in 2022 and a fourth-rounder in 2023 in the deal.
Michel fills an immediate need for the Rams, who will play the season without rising star Cam Akers after he tore his right Achilles tendon while working out before training camp.
In Akers' absence, Darrell Henderson Jr. has taken over as starter, but the Rams lack depth behind the third-year pro.
"We identified Sony for many reasons," Rams general manager Les Snead said Wednesday. "Number one, being his experience, and experience meaning like a player who has carried the ball in big games and in those big games, where there have been big moments, big drives, big series."
At the outset of training camp, coach Sean McVay said the Rams would give their young backs ample opportunity to compete for the job but left open the possibility that the team could look for outside help when the season opener approached.
Asked this week whether Jones and Funk had proved through two preseason games that they could be ready for Week 1 against the Chicago Bears on Sept. 12, McVay said the Rams were in the process of figuring that out.
"We got a little bit of time," McVay said. "Darrell is a guy that's been established and done some good things. But I was pleased with both those guys."
Henderson suffered a mild thumb sprain in practice Monday but returned to practice Tuesday with a cast or brace protecting his left hand.
A fourth-year pro, Michel has rushed for 2,292 yards and 14 touchdowns in 28 starts for the Patriots, who drafted him in the first round with the 31st pick in 2018.
"What he does bring is this element of make a cut, get north and south and get north and south with some force," Snead said. "The physics that he brings to the table -- let's call it a flavor or genre that we felt like complemented our group."
In a 13-3 victory over the Rams in Super Bowl LIII, Michel rushed for 94 yards and a touchdown on 18 carries.
Michel is due to earn a base salary of $1.79 million in the final year of his contract in 2021 after the Patriots previously elected not to exercise his fifth-year option for 2022. He had drawn rave reviews from Patriots running back coach Ivan Fears this preseason, with Fears saying he had "upped his total game."
"I've always liked him as a runner, but now he's shown to us that he can also play in the passing game and be very effective," Fears said. "He's had an awfully good camp."
Last season, Michel missed six games when he was placed on injured reserve after Week 3 because of a quadriceps injury. In nine games he rushed for 449 yards and a touchdown on 79 carries. In a 24-3 loss to the Rams, Michel rushed for 22 yards on seven carries.
The Patriots have gone through training camp with a surplus of six running backs, including third-year pro Damien Harris, who took over the starting job from Michel last season, and veteran James White.
The Patriots' surplus at running back has been a hot topic at camp, with Harris, Michel, White, fourth-round pick Rhamondre Stevenson, spark plug J.J. Taylor and veteran core special teams player Brandon Bolden on the depth chart, and the likelihood that the team would keep a maximum of five players at the position.
Last week, coach Bill Belichick said, "They have all gotten opportunities and shown positively. Competition brings out the best in all of us. It's a great situation."
ESPN's Mike Reiss contributed to this report.