Wednesday had the majority of chances to score but they failed to capitalise on them and Bowen's second-half goal was enough to give Grant McCann's men all three points as they made it eight wins out of their last 14 Championship outings.
The first effort of the match came from Wednesday's Barry Bannan, who volleyed wide.
Grosicki then put a free-kick on target but it was comfortably saved by Cameron Dawson.
Adam Reach wasted a good opening for the home side when he got on the end of a Harris low cross, failing to make a proper connection and firing wide.
Pennington then came to Hull's rescue, clearing a shot from Harris off the line.
The Wednesday line-up showed a change in attack at the start of the second half with Fletcher taking over from Nuhiu.
And it was a move that nearly paid dividends as Fletcher was soon in the thick of the action, with his effort hitting the post after he got on the end of a Bannan corner.
Monk made another attacking change just before their hour-mark, sending on Jordan Rhodes in place of Winnall.
But Bowen struck after 61 minutes to put the visitors in front, drilling a hard, low shot past Dawson after a good ball in from George Honeyman.
Grosicki threatened to add a second shortly afterwards, firing over from inside the area.
Fletcher fired narrowly wide after winning a free-kick inside the 'D' and Rhodes put a near post header beyond the far post as Wednesday looked to draw level.
Murphy played a searching ball into the area in the closing stages of the match but it just evaded Fletcher who was at full stretch in the middle.