Overton, 31, suffered the injury after bowling his second ball of England's eventual 238-run win, as he spilled a fast-travelling return catch off Keacy Carty and immediately sprinted off the field for treatment.
Jacob Bethell completed Overton's over, though he did return to the field later in the innings, with his digit heavily strapped, and claimed three late wickets to finish with England's best figures of 3 for 22 in 5.2 overs. The injury appeared at the time to be a dislocation, but it has since been confirmed as a fracture.
During his spell he dismissed Roston Chase and Matthew Forde with short-pitched deliveries and deceived Gudakesh Motie with a slower ball.
Overton will now undergo a period of rehabilitation under the supervision of the England medical team, but no replacement will be added to the ODI squad. Matthew Potts and Luke Wood, the Lancashire left-arm seamer, are the reserve quicks in contention, while the left-arm spinner Tom Hartley could also come into consideration.
The second of the three ODIs takes place in Cardiff on Sunday, with the series concluding at the Kia Oval on Tuesday before the first T20I gets underway in Chester-le-Street on Friday.