After Matthew Virtue fired Liverpool's best first-half chance straight at West Brom goalkeeper Alex Palmer, the home side went ahead shortly before the break.
Morton (pictured) was fouled by Liverpool goalkeeper Shamal George and scored the resulting penalty.
The Reds pushed for a leveller in the second half but Palmer kept out Isaac Christie-Davis's free-kick before Virtue and Fidel O'Rourke went close.
Nottingham Forest earned a 3-1 Group F success against Exeter City at the Impact Arena.
Toby Edser struck Forest's 23rd-minute opener before the home side's goalkeeper Liam Bossin made a superb save from Jack Sparkes to deny Exeter an equaliser.
Jayden Richardson doubled Forest's advantage on 39 minutes before substitute Yassine En-Neyah made it 3-0 shortly after coming on.
James Dodd pulled a goal back for Exeter on 64 minutes but the home side held on to go level on points with Leicester City at the top of Group D.