It secured the club’s first piece of silverware since the 1980 FA Cup and their first European trophy since 1965.
Bowen told BT Sport: “You always say you want to score in the last minute and to do it here in front of these fans, I thought I was going to cry.
“I’m so happy. We haven’t had the best season, myself included, but to do it tonight, I’m over the moon.
“I think in my position you make that run 10 times you might get that ball once. As soon as you get it you’ve got to put it away.”
Bowen had earlier won the penalty from which Said Benrahma opened the scoring but Giacomo Bonaventura equalised just four minutes later for the Serie A side.
But with the game appearing set to head to extra-time, Bowen struck his 13th goal of the campaign and earned West Ham a place in next season’s Europa League.
The England international added: “Never (have I felt like this in my life). This is the biggest game of my career. The emotion, there was time for one more chance. I’m just so happy. I’m over the moon.
“I’m thinking of the party tonight. Listen to it. Listen.”