The difference in the game was the Dolphins' ability to get into the end zone, on Ronnie Brown's 3-yard run in the first half, and winning the turnover battle. While Goodman, Akin Ayodele and Renaldo Hill all recorded interceptions for the Dolphins, Miami's only takeaway came on a David Martin fumble.
Chad Pennington had modest numbers in this game, passing for 166 yards with no touchdowns or interceptions, but the running game contributed 149 yards.
The Rams welcomed back Steven Jackson on Sunday, and he rushed for 94 yards on 21 carries.
It wasn't a dominating performance by any stretch by the Dolphins defense, but they did force the Rams to settle for a field goal on all three trips inside the 20-yard line.
The first came on the first drive of the game when two pass interference penalties gave the Rams a first-and-goal at the 7.
After two Josh Brown field goals gave St. Louis a 6-0 lead, the Dolphins put together the only touchdown drive of the game after a holding penalty on a kickoff return forced them to start at their own 9-yard line.
They also overcame a facemask penalty on Vernon Carey that put them in a first-and-22 hole at the St. Louis 38.
The drive began with runs of 10, 12 and 10 yards by Lousaka Polite, Patrick Cobbs and Ricky Williams, and was kept alive later by completions of 13 yards to Ted Ginn Jr. and 19 yards to rookie Davone Bess.
Starting in place of the injured Greg Camarillo, Bess led all receivers with six catches for 84 yards.
St. Louis' last lead of the game came at 9-7, but the Dolphins answered with a field goal drive to close out the first half.
The Dolphins had an apparent touchdown run by Williams nullified by a holding penalty on guard Ikechuku Ndukwe and eventually had to settle for Dan Carpenter's 37-yard field goal.
Carpenter added field goals of 47 and 42 yards, sandwiched around a 38-yard by Brown, to make it 16-12 heading into the final minutes.
The Dolphins could have closed out the victory with another first down, but Coach Tony Sparano took the conservative approach and ran a pitch to Ronnie Brown on third-and-13 from the Miami 41.
The Rams began their final drive at their 25 and moved to the 47 before Goodman picked off Bulger's deep pass down the left sideline.
The victory moved the Dolphins to 7-5 and kept them in the thick of the AFC playoff race. They travel to Toronto next Sunday to take on the Buffalo Bills in the first regular season game played in Canada.