Overton, 31, suffered the injury while attempting to take a return catch off Keacy Carty just two deliveries into his opening spell. He left the field immediately for treatment and, though he later returned with his finger heavily strapped, subsequent scans confirmed a fracture rather than a dislocation.
Despite the setback, Overton made a strong impact in England's 238-run victory, finishing with figures of 3 for 22 in 5.2 overs. His dismissals included Roston Chase and Matthew Forde with short-pitched deliveries, as well as Gudakesh Motie, who fell to a well-disguised slower ball.
He will now begin a period of rehabilitation under the supervision of the ECB medical staff. England have opted not to name a replacement, with Matthew Potts and left-armer Luke Wood already in the squad as reserve seam options. Left-arm spinner Tom Hartley may also come into contention for selection.
The second ODI will be played in Cardiff on Sunday, with the final match of the series scheduled for Tuesday at The Kia Oval. The T20I leg of the tour begins on Friday at Chester-le-Street.