All five goals at Portman Road came in the second half, the first almost immediately from the restart from Aaron Drinan for the hosts.
On 53 minutes, Ipswich's Sam Goteni was sent off for a foul in the area on Sam Field, who equalised from the penalty spot.
Ben Morris's stunning finish restored Town's lead but West Brom pegged them back again through Jamie Soule's volley.
Ipswich gave the Baggies no time to recover for a third time when Drinan won it with his second in the 90th minute.
West Brom: 5 Mar v Reading (A) PL2