Sule saw a chance go begging before creating another brilliant opportunity for McNeil, but his finish flew over when well placed to test the goalkeeper.
Frustration boiled over as the visitors started to fade, and it saw the Rs reduced to 10 as the game entered its closing stages – Isak Alemayehu picking up a second yellow card to make the Londoners’ task even harder.
Albion should have been out sight come the 88th minute but for a brilliant save from Charlie Warren to deny Souleyman Mandey at close range, before the Baggies man saw another goalbound effort blocked moments later.
But the hosts weren’t made to pay for their lack of a second strike – and cruised through the seven minutes of added time to ensure they booked their place in the quarter-finals.
Albion: Cisse, Maughan, Diomande (c), DuPont, Nelson, Mfuamba, Colesby, Mandey, McNeil (Ranger 90+4), Ntege, Sule (Chimeziri 88).
Subs not used: Okonkwo, Round, McDonald.