Smith's leadoff blast to right field off Anthony Swarzak (2-1) came after the Padres tied the score in the top of the ninth on Everth Cabrera's sacrifice fly.
San Diego's Alexi Amarista then made game-saving over-the-shoulder catch in center field on Eduardo Escobar's blast in the bottom of the ninth with two runners on base.
The Padres snapped a nine-game losing streak to the Twins, which included a 3-1 setback on Tuesday in the opener of the two-game series. It was their first win over Minnesota since June 19, 2005.
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