The Hammers faced a monumental task of ending the German side's season-long unbeaten run to give themselves any chance of progression.
And for 76 minutes, it looked like they would be the first team to beat Leverkusen in 44 attempts after Michail Antonio's 13th-minute header gave the hosts hope.
The Hammers flew out of the blocks like they knew the exact scale of their task - to beat Bayer Leverkusen for the first time since May 2023.
It set the tone for a one-sided first half on and off the pitch, as Alonso's much-changed champions struggled for rhythm or intensity.
Even so, it was Leverkusen who first threatened when Nathan Tella's long-range drive was turned behind, but moments later West Ham had the lead on the night.
Bowen's turn and curling cross were the perfect assist for Antonio, who beat Odilon Kossounou with ease in the air before nodding past a helpless Matej Kovar.
Leverkusen's start was so poor the Ivory Coast defender was hooked with less than half an hour played, but not before Mohammed Kudus, Antonio and Bowen had all come close to bringing the tie level on aggregate.
That change provided some measure of strength for the visitors at the back, but Ward-Prowse was still able to fire into the body of Kovar after Julian Alvarez's effort was blocked moments before the break.
Another two changes at the interval hinted at what West Ham had feared, a reinvigorated Leverkusen. And after a presumed Alonso hairdryer treatment, they finally carried the swagger of Germany's best team.
Even so, West Ham kept them at arm's length for the most-part.
Frimpong's snapshot from a near impossible angle was pushed away by Fabianski but it was not until the Hammers began to push forward for the all-important second that they truly threatened.
Frimpong blazed over when played in behind and Florian Wirtz sliced a half-volley horribly wide, while Bowen wasted West Ham's one chance of note - and perhaps their best of the game.
Harrying Hincapie off-balance and robbing the defender in his own box, he had three men up in support but did not look up and flashed a shot far beyond all three with little chance of beating Kovar from a tight angle.
Hope still remained for the Hammers until the final minute, when Frimpong brought down a ball in the box and a significant deflection off Cresswell ended in the back of the net to finally end their valiant challenge.
Thousands of West Ham fans chose that moment to leave but greeted their side back to the centre circle for kick-off with a round of applause to show their appreciation of running one of Europe's best sides so close. Just not quite close enough.