Alpine said he would start next week, ahead of the Azerbaijan Grand Prix in Baku.
Elliott was described by principal Toto Wolff as a 'pillar of the team's achievements' when he left Mercedes in October 2023 after an 11-year stint during which they won a record eight successive constructors' titles and seven drivers' championships.
Alpine, sixth in the standings and nine points behind fifth-placed Racing Bulls, said Elliott would lead and oversee the technical and engineering functions at the Enstone factory.
The Briton previously worked as principal aerodynamicist at Enstone from 2008 to 2012.