The burly young striker - captured on loan from Arsenal for the rest of the season - broke the deadlock with a stunning strike to set the Baggies on their way a fourth round tie against Burnley.
Albion knocked Posh out of two cup competitions last season, although hopes of a giantkilling were high after the League One promotion chasers shared the spoils when the teams renewed acquaintances at the Hawthorns ten days ago.
But Simpson soon silenced the shock-seekers in a bumper 10,735 crowd by steering a 25-yard curler into the bottom corner of the Posh net after 18 minutes
Home keeper Joe Lewis had already saved from Roman Bednar in a one-on-one situation and defender Leon Barnett had squandered a golden headed chance before Simpson struck in some style.
Albion continued to dominate the opening period and deservedly doubled their advantage eight minutes before the break when Paul Robinson was left unmarked to head home a pinpoint Kim Do-Heon cross from the right.
Simpson was denied a second goal by a smart Lewis save early in the second period before Kim was only a whisker away with a long range thunderbolt.
At the other end Posh rarely looked like mounting a revival.
Charlie Lee failed to convert the home side's best chance after only nine minutes when lifting the ball over Albion keeper Scott Carson only for it to drop inches wide.
Carson made a hash of blocking another Lee drive after a lung-busting 65th minute surge, before a last-ditch stoppage time tackle from central defender Ryan Donk prevented Aaron McLean from tucking away a consolation.