The showpiece event appeared to be heading for extra time after Said Benrahma's converted penalty kick on the hour mark was cancelled out just seven minutes later by Giacomo Bonaventura.
However, the Hammers snatched victory in the 90th minute when Bowen slotted a composed strike beyond La Viola goalkeeper Pietro Terracciano after receiving a perfectly-weighted through-ball from Lucas Paqueta, sparking pandemonium among the travelling Hammers supporters.
Following a tepid domestic campaign that saw them narrowly avoid relegation from the Premier League, West Ham held on to clinch their first major trophy since the 1980 FA Cup and their first European silverware for 58 years.
Bowen, who has ended the season with five goals and three assists in 11 ECL matches, was playing non-league football with Hereford United less than 10 years ago and has now etched his name into West Ham folklore after scoring the decisive goal on a historic night in a major European final.
Indeed, the attacker is the first Englishman to score a winning goal in a major European final since Alan Smith for Arsenal in the 1994 Cup Winners' Cup final against Parma.
Speaking to BT Sport after the match, an emotional Bowen said: "I obviously dreamed of scoring, but to score the winner in the last minute - It's what you always say you want to do. To do it in front of these fans. I thought I was going to cry. I'm just happy, I can't put it into words.
"We had a dream, we haven't had the best season, myself included, but to do what we've done tonight for these fans - we've said before, we see it all on Instagram, all the fans. To give them this moment, I'm over the moon."
The 26-year-old added: "I said leading up to this, it's the biggest game of my career. The emotion, I thought it was going to go to extra time but there was time for one more chance.
"I'm just so happy that we won it for everyone. Look at everyone, it means so much. I'm over the moon."
Meanwhile, David Moyes has described West Ham's Europa Conference League triumph as the best moment of his 25-year managerial career.
The 60-year-old, who endured intense scrutiny earlier this season amid fears of relegation from the Premier League, marked his 1,097th match in management by winning his first-ever major trophy and was able to revel on the pitch with his 87-year-old dad in front of the Hammers faithful.
"This is fantastic. I have had a long career in football and you don't get many moments like this," Moyes told BT Sport. "These sorts of moments as managers don't come around often. This is a great moment for us.
"It is a brilliant club in the East End of London that does wonderful work in the community. It is a great family club. It is getting better and stronger. This is another step on the road of continued progress.
"We have gone unbeaten in Europe, which is incredible. We only lost last year in a semi-final, two incredible years and we get another one now."
Asked if winning the Europa Conference League is the best moment of his career, Moyes said: "I would have to say so. The moments you get to celebrate with your family and win in the last minute of the game, it doesn't happen often. It can go against you but tonight is a brilliant feeling.
"If somebody had said three years ago when I took the job that you'd avoid relegation and finish in Europe and I've said you were mad. This competition has been great for us, the players have been remarkable."
Meanwhile, captain Declan Rice has acknowledged the speculation surrounding his future at West Ham and has confirmed that there is "interest from other clubs" in his services ahead of the summer transfer window.