Leach accounted for Richard Levi and Rob Keogh, both caught behind, before removing Ben Duckett, caught at gully.
The 24-year-old returned with figures of 4-30 but still ended up on the losing side as Worcestershire fell short of their modest 126-run chase by 21 runs.
Leach though is not the first bowler to achieve the remarkable feat in cricket history.
At the 2003 World Cup, Sri Lanka's Chaminda Vaas became the first player in Tests or one-day internationals to take such a hat-trick, in a win against Bangladesh. He went on to claim a fourth victim in the opening over itself.
Northants' second win of the competition leaves Worcestershire on one point from four games in Group A.