The Gunners began the game just a point ahead of fifth-placed Manchester United and with their hopes of qualifying for the Champions League for a 19th season in a row seemingly at risk.
The club's supporters had seemingly chosen to vote with their feet, with the half-full Emirates Stadium littered with empty seats for the clash.
However, Sanchez struck early on from the edge of the area and added a second from a free-kick seven minutes before half-time as Arsene Wenger's side returned to winning ways after back-to-back draws.
Gareth McAuley's header came off the crossbar in the closest the Baggies came to finding the net, but Tony Pulis' side slipped to a third defeat in a row.
The Gunners have now moved back to within five points of rivals Tottenham as they look to overhaul Mauricio Pochettino's side in the final four games of the season.