In front of a deafening 48,000 crowd, Rafael Borre drilled in the winner for the hosts in the 26th minute, beating the English club for a second time following a 2-1 triumph in the first leg last week.
Their manager David Moyes was also sent off in the 79th minute for angrily kicking a ball in a frustrating and scrappy end to West Ham's European run this season.
"A dream has finally come true," Eintracht president Peter Fischer said, surrounded by ecstatic fans who stormed the pitch on the final whistle. "The team did it really well. This city and this environment deserved it. That's football and now we will win this thing."
It had not started well for Frankfurt, though, with defender Martin Hinteregger taken off only eight minutes with a hamstring injury.
But the dismissal of Creswell opened up space down the wing for the Bundesliga side and Ansgar Knauff made the most of it when he charged through and delivered a perfect ball for Borre to score.
West Ham hardly got a look in and their best chance came a minute from half-time when Kurt Zouma's close-range effort was cleared off the line.
They were equally ineffectual after the break with Moyes's frustration boiling over when he kicked the ball at a ball boy who he thought was trying to waste time.
"Really disappointed," Moyes said. "Things didn't go for us tonight. I am really proud of the players -- how they played with 10 men was fantastic. Other teams would have folded and lost 2-0 or 3-0.
"I felt this was a chance. I think we've played better teams than Frankfurt. If we're honest, we probably lost the tie in the first few seconds at the London Stadium."
With the home fans kicking off celebrations early for what turned out to be a memorable evening for the club struggling mid-table in the Bundesliga, Eintracht comfortably held on to their lead.