Stafford overcame a generally poor game in which he was intercepted three times to pull out the late win at home in the season opener for Detroit (1-0), leading the Lions to a pair of fourth-quarter touchdowns.
Smith compiled 62 rushing yards with a score and made four receptions for 29 yards and the winning TD for the Lions.
The Rams (0-1) were led by Sam Bradford's 17-for-25 passing performance for 198 yards and a score.
Brandon Gibson paced the St. Louis receiving corps with four catches for 51 yards and a touchdown.
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