Benin forward Sessegnon, 29, was surplus to requirements after Sunderland signed Fabio Borini on loan.
He joined Sunderland from Paris St-Germain in 2011 and West Brom say his fee surpasses the reported £6.5m they paid for Shane Long.
Anichebe, 25, has spent his career so far at Everton.
The Nigeria international has scored 26 goals in 168 appearances for the Toffees.
West Brom have signed Sunderland's Stephane Sessegnon for a club-record fee and Everton striker Victor Anichebe for a fee which could rise to £6m.