Ramsdale was omitted from United's squad for the Sky Bet Championship game as the transfer neared completion after the clubs agreed an initial fee of around £24 million.
Ramsdale's place went to Michael Verrips, who endured a torrid time, allowing a corner to slip through his hands for Albion's second goal then beaten in the air for the fourth as West Brom went top of the table after three games.
Valerien Ismael's side continued their bright early form and had the ball in the net twice in the first 16 minutes, but both efforts from the former Sheffield United forward Callum Robinson were ruled out.
Robinson prodded over the line after 64 seconds when Matt Clarke flicked on a long throw-in by Darnell Furlong, but Robinson was judged to be offside and the ball hit him on the arm.
Then Robinson volleyed home after a Furlong flick-on but this time there was a foul in the build-up.
West Brom's pressure eventually told with a goal that did count, in the 26th minute.
Yet another long throw-in from Furlong saw a cluster of players from both sides rise on the edge of the six-yard box and the ball flew in off the unfortunate head of United left-back Jack Robinson.
Almost immediately, United should have equalised but David McGoldrick sidefooted wide with only goalkeeper Sam Johnstone to beat.
West Brom killed the game off with three goals in 12 minutes before the hour mark.
They doubled their lead in the 47th minute - and it was one poor Verrips will want to forget.
The 24-year-old Dutchman first flapped with one hand then allowed Alex Mowatt's corner to squirm through both hands, with Dara O'Shea gleefully poking home through a crowd of players.
Six minutes later it was 3-0 as Albion sliced through United with consummate ease.
Mowatt started and finished the move. He won the ball in the air, finding Callum Robinson and raced for the return off Karlon Grant before coolly sidefooting past Verrips.
Callum Robinson finally got his name on the scoresheet in the 59th minute, controlling and volleying home from close range after another Furlong throw-in was flicked on by Kyle Bartley, with Verrips conspicuous by his absence.