With the scored tied 16-16 and 3:13 remaining, the veteran running back scored on a 15-yard pass from quarterback Matt Cassel.
Faulk finished with 60 yards on 13 rushes and added 47 yards on four receptions.
Wide receivers Randy Moss and Wes Welker combined to make 14 catches for 181 yards as Cassel completed 21-of-33 passes for 267 yards, one touchdown and two interceptions for the Patriots, who have won three of four.
St. Louis' Marc Bulger was 18-for-34 for 301 yards, one touchdown and one interception.
Antonio Pittman, in the absence of Stephen Jackson, rushed for 83 yards on 19 attempts.
Donnie Avery scored once on six catches for 163 yards for the Rams (2-5), who suffered their first loss under interim Coach Jim Haslett.
Kickers Josh Brown and Stephen Gostkowski hit three field goals each.
Notable deaths of 2014 [PHOTOS]