Hurts, the Eagles quarterback and 2022 MVP runner-up, threw for 193 yards and two touchdowns.
His running-back Swift carried for 175 yards and crossed for a touchdown himself.
The Eagles jumped out to a 27-7 lead in the third quarter after a mistake-prone game from the Vikings.
However, Minnesota fought their way back into the game with two touchdowns in the last two quarters to make it 27-21.
A late touchdown from Swift with four minutes left made the comeback insurmountable despite a late score from Vikings tight-end TJ Hockenson.
Vikings quarterback Kirk Cousins threw for 364 yards with four touchdowns while wide receiver Justin Jefferson had 11 receptions totalling 159 yards.
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The performance from the duo was not enough to give their team their first win of the season, with the Vikings to host the Los Angeles Chargers next week while Philadelphia will travel to Florida to face the Tampa Bay Buccaneers.
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