Smith also threw a 21-yard touchdown pass to Obafemi Ayanbadejo, while Josh McCown threw for 157 yards on 12-of-18 passing for the Cardinals (1-3).
Troy Hambrick ran for 79 yards and a score for Arizona.
Aaron Brooks completed 24 of his 40 passes for 242 yards for New Orleans (2-2), which had won six of the last eight meetings with Arizona.
Mel Mitchell recovered a blocked punt in the end zone and Aaron Stecker had 23 yards rushing and 71 yards receiving for New Orleans.
Smith, the NFL's all-time leading rusher, fumbled inside the New Orleans five-yard line but redeemed himself when he threw the TD pass for a 14-3 lead with 1:12 left before halftime.