Bowen wrapped up a comfortable 2-0 win for the Hammers with a breakaway goal after Manuel Lanzini’s first-half strike put them in control.
The 25-year-old is one of West Ham’s stand-out performers this season and is sure to be on the radar of Three Lions boss Gareth Southgate.
“If he keeps playing at the level he’s playing at he’s going to get closer,” said Moyes.
“The competition is so strong in that position, but if you’ve got wide players who score goals and make assists he’s got a chance.
“Hopefully he’ll keep that pressure on Gareth. As a manager you enjoy the thought of helping someone become an international.
“We’ll keep pushing Jarrod, keep trying to get him scoring and improve his game, and hopefully get him an England cap.”
Bowen had an effort cleared off the line and a shot saved by Illan Meslier before he got the goal his display deserved.
With Meslier having to run back after going up for a last-ditch corner, Michail Antonio broke clear and squared for Bowen to tuck in his sixth goal of the season.
Bowen also played a part in the opening goal, with the VAR painstakingly checking if the frontman, in an offside position, had touched the ball before it landed at the feet of Lanzini.
Leeds head coach Marcelo Bielsa refused to be drawn on any hint of controversy, saying: “It’s a decision I prefer to accept what the referees say.
“Not because I haven’t created an opinion, but it’s a collaboration us managers should make, (we) should accept their decisions even if they jeopardise us.”
Defeat condemned Leeds to a ninth FA Cup third-round exit in the last 12 seasons.
Bielsa added: “I think it corresponded that they won the game. They created more danger than we did. They had balls with which to attack in worse situations than we did and created more danger than us.”
Bielsa also revealed Patrick Bamford had been due to play but picked up a hip injury in training.
Moyes complained that West Ham had been getting tough draws in the cups, having beaten Manchester City and Manchester United in the Carabao Cup this season before going out to Tottenham.
Moments later the Hammers landed a fourth-round trip to non-league Kidderminster.
He said: “I want to try and go far in the cup. This was probably one of the toughest ties in the draw. I’m really pleased we kept a clean sheet because Leeds cause you problems.”
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