Craig Dawson's double piled more pressure on Wenger as the Gunners suffered a nightmare 3-1 defeat at The Hawthorns.
The defender and substitute Hal Robson-Kanu, who scored just 75 seconds after coming on, inflicted a sixth loss on the beleaguered Wenger in his last nine games.
Alexis Sanchez's 18th Premier League goal of the season had cancelled out Dawson's opener, but the Gunners remain five points adrift of the top four with their hopes of Champions League qualification in the balance.
Both of Dawson's goals came from simple corner routines, while David Ospina - on for the injured Petr Cech - failed to cut out a high ball as Robson-Kanu stole in to finish the rebound to seal the game.
Former Tottenham and Liverpool midfielder Redknapp told Sky Sports: "That’s the sort of performance that gets managers the sack.
"That was shambolic from Arsenal. I have never seen a team at the top level, defending against a team we know are great from set-pieces, defend like that.
"Surely somebody has told them before the game you need to be very careful, McAuley, Dawson, they are monsters at set-pieces.
"You need leaders in the dressing room, somebody has to be organising, but it was a parting of the ways.
"I have not seen such shambolic defending. I feel so sorry for those Arsenal fans that have come up in their droves. I thought there would be a performance but that was diabolical.
"Arsene Wenger has taken a lot of stick, I hope he’s in that dressing room telling those players exactly what he thinks of them because they have let him down and let the club down."