A penalty shootout was looming at the London Stadium until England forward Bowen found the net two minutes from the end.
A shot from substitute Mohammed Kudus hit the chest of Bowen and flew past Cherries goalkeeper Neto to send the Hammers through.
The final touch may have come off Bowen's elbow, which will no doubt stick in the craw of Bournemouth who had a stoppage-time goal disallowed by VAR against Newcastle for handball on Sunday.
With no VAR in operation in the early rounds of this competition, the goal stood.
It ended a contest low on quality despite West Ham handing full debuts to almost £100million-worth of summer signings.
Aaron Wan-Bissaka, Jean-Clair Todibo, Niclas Fullkrug and Crysencio Summerville all started for the hosts.
But it was only the impressive Wan-Bissaka, the £15m recruit from Manchester United, who was still on at the end after a distinctly mixed bag of performances.
Bournemouth almost opened the scoring when Marcos Senesi headed a corner back across goal and James Hill planted his header narrowly wide.
West Ham ended the first half strongly with Wan-Bissaka, playing at left-back after Aaron Cresswell dropped out of the starting lineup, prominent in attack.
The former Crystal Palace full-back cut the ball back across goal but it took a deflection to wrong-foot Fullkrug, who was lurking six yards out.
Moments later Fullkrug collected another pass from Wan-Bissaka and saw his shot deflected wide for a corner, from which Tomas Soucek scooped a good chance over.
Summerville then ran at the Bournemouth defence for the first time but got underneath the ball and his finish flew horribly off target.
Bournemouth almost scored in first-half stoppage time when Hill's cross was cleverly backheeled towards goal by Dango Ouattara, forcing a smart reaction save from Lukasz Fabianski.
After the break Bowen found Summerville, who cut inside Hill and bent his shot inches wide of the far post.
Back came Bournemouth, with Fabianski saving low down to deny Ouattara and their £42m record signing Evanilson, on as a substitute, sliced a gilt-edged chance wide.
They were agonisingly close to breaking the deadlock when Marcus Tavernier's curling free-kick from out wide grazed the post.
Hammers captain Bowen forced a first save from Neto after 80 minutes with a low drive from the edge of the area.
But they broke the deadlock when Bowen found Lucas Paqueta, continued his run forward and bundled Kudus' shot home.
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