The 25-year-old scored his fourth and fifth Premier League goals of the season, had another denied and was twice denied by the woodwork as he caused constant problems for Norwich at London Stadium.
David Moyes’ side are up to fourth in the table, and while Arsenal and Tottenham below them have games in hand, the Hammers look confident after four straight wins in all competitions.
West Ham nearly took the lead within two minutes of a quick start, but Bowen’s shot was saved well by Tim Krul.
Milot Rashica twice found space at the edge of the box for Norwich, only to produce a controlled effort on goal, while Manuel Lanzini was equally useless when offered chances in promising positions .
It looked like West Ham had taken the lead after 34 minutes, when Bowen’s cross from the right beat everyone and found the far corner. However, that was ruled out when reruns showed that Issa Diop and Nikola Vlasic, both in an offside position, attempted to get to the end in front of Krul.
Bowen was not to deny, and the 25-year-old finally scored his goal a few minutes before the break. Vladimir Coufal threw a nice ball into the box and Bowen found space six yards to get home.
The West Ham winger continued to be a constant threat after the break and had the bad luck not to add to his record.
Pablo Fornals played it with a superb pass, and Bowen beat Krul with a nice dink but saw his effort bounce off the bar. He was again refused by the woodwork, as Brandon Williams deflected a shot to the outside of the post.
Adam Idah nearly leveled things off after 67 minutes. He showed excellent footwork to find space 20 yards before unleashing a fierce strike, which Lukasz Fabianski knocked down on the post.
The game grew increasingly nervous as home fans feared a late punch from Norwich.
But Bowen put the match to bed late, having returned Arthur Masuaku’s cross at close range. He was initially ruled out for offside, but VAR was his friend on this occasion as the goal was awarded after reruns showed he was onside.
Andriy Yarmolenko saved a shot as West Ham took advantage of the final minutes knowing they were coming back to the Champions League places.