Nuno was appointed Hammers manager on Saturday, tasked with turning around a side that had started the Premier League season with four defeats in five matches, leading to Graham Potter's sacking.
West Ham's dire run looked set to continue when they fell behind early on to Michael Keane's header but while Everton hunted a second after the break, Bowen struck to guarantee the visitors a point.
As well as being the first goal of Nuno's reign, Bowen became the first person from a visiting team to score at Everton's new Hill Dickinson Stadium, denying Toffees boss David Moyes a win over his former side.
Moyes' side started brightly and exploited West Ham's set-piece frailties to go ahead in the 18th minute.
Under Potter, the Hammers had conceded seven times in the league from corners this season and that weakness shows no sign of disappearing any time soon.
While Alphonse Areola punched the initial delivery clear, the ball was worked backed to James Garner, whose cross was headed into the net by an unmarked Keane.
Chants of 'sack the board' and 'we want our club back' soon rang around the away end and matters might have worsened before halftime as Kiernan Dewsbury-Hall glanced a header wide of the far post.
Everton went up a gear immediately after the break as Jake O'Brien and Idrissa Gueye rifled over the bar, while Jack Grealish teed up Garner, who was twice denied from the edge of the area by Areola.
Grealish's appeals for a penalty fell on deaf ears when his low cross ricocheted off Areola and cannoned into the left hand of an oblivious Konstantinos Mavropanos.
There was no denying West Ham on their next foray forward, though, with El Hadji Malick Diouf sprinting clear of O'Brien and James Tarkowski and sending a cross to the far side via Keane's attempted clearance.
Bowen collected the loose ball on the edge of the box, cut on to his left foot and arrowed a fantastic shot - which brushed Keane's head - beyond the helpless Jordan Pickford in the 65th minute.
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