Lanzini’s first-half goal and Bowen’s late strike secured a 2-0 win for David Moyes’ side and condemned Leeds to a ninth third-round defeat in the last 12 seasons.
It is now five years since the Whites have won an FA Cup match, and they rarely looked like bucking that sorry trend in a one-sided London Stadium encounter.
Moyes named a strong Hammers side and they almost took the lead after just 10 minutes when Declan Rice’s cross from the byline was met with a poor defensive header.
The loose ball fell at the feet of Bowen but Luke Ayling had managed to get back onto the goal-line to block the forward’s shot.
A host of injuries in the Leeds squad had prompted boss Marcelo Bielsa to hand debuts to Lewis Bate and Leo Hjelde and a first start to a third 19-year-old, Sam Greenwood.
The hosts were dominating, though, and their first goal arrived in the 34th minute when Nikola Vlasic tried to give Hjelde the slip in the area and both landed in a heap.
Vlasic managed to hook the ball away from Leeds keeper Illian Meslier and it eventually squirmed back to Lanzini, who lashed it home from 10 yards.
VAR Peter Bankes had a long, hard look at the replays before eventually deciding there was no offside against the lurking Bowen, and the goal stood.
Ryan Fredericks should have doubled the lead in first-half stoppage time after a lovely one-two with Bowen, but the full-back’s finish was too close to Meslier.
Bielsa replaced Greenwood and Bate with Raphinha and Stuart Dallas at half-time.
But it was West Ham who were the more threatening after the interval with Meslier saving with his feet from Bowen and Fredericks fizzing a shot across goal and wide.
Instead another early Leeds exit was confirmed when Antonio broke clear from a visitors corner and squared for Bowen to wrap up the victory.
Register for free to continue reading
Registration is a free and easy way to support our truly independent journalism
By registering, you will also enjoy limited access to Premium articles, exclusive newsletters, commenting, and virtual events with our leading journalists
Already have an account? sign in
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies