The forward's brace ended the Magpies' month-long goal drought before Valentino Lazaro added a third after the break.
It is the first time under owner Mike Ashley Newcastle have reached the last eight of the competition.
The impressive Allan Saint-Maximin also hit the post for the Magpies who dominated the Baggies, top of the Sky Bet Championship.
Two of Newcastle's previous three wins in 2020 came against Rochdale and Oxford in the FA Cup and Bruce's side continued their form in the competition.
Nabil Bentaleb, Almiron and Saint-Maximin were all guilty of failing to find the final pass but the visitors settled quickly.
Sean Longstaff lashed a wild half volley over after 22 minutes and Saint-Maximin went closer six minutes later.
The returning forward collected Joelinton's pass to advance and smash a wonderful 25-yard effort off the post.
Two minutes later a rare Albion attack saw Darlow turn Charlie Austin's drive over but Newcastle soon got the goal they deserved.
The Magpies had not scored in a month – since their fourth round extra-time win at Oxford – but Almiron ended the wait after 33 minutes.
Saint-Maximin had been causing problems and spotted Almiron's run from deep, which remained unchecked by the Baggies.
The Paraguay international was allowed to collect the pass and finished low past Bond.
The Magpies deserved the lead and doubled their advantage in first-half stoppage time.
Their front three combined with a clinical edge they have failed to show in the the Premier League when Saint-Maximin fed Joelinton and his cute backheel was lashed in from 12 yards by Almiron.
The Magpies seemingly ended the contest 75 seconds into the second half when Bond palmed Saint-Maximin's cross straight into Lazaro who could not miss.
Albion boss Slaven Bilic had made nine changes in a clear sign of his priorities as the Baggies chase promotion back to the Premier League.
He would have been pleased with his side's response though and Filip Krovinovic shot over.
Krovinovic's 57th minute introduction had given Albion extra impetus and the hosts made it 3-1 after 74 minutes.
Edwards struck the bar with a deflected 20-yard shot and, after Newcastle failed to clear their lines from the corner, Phillips eventually smashed in off the turf.
Zohore made it 3-2 in injury time, finishing neatly at the near post, but it came too late to derail Newcastle.