Eintracht Frankfurt clinched a place in the Europa League final and ended West Ham's dream run as Rafael Borre netted in a 1-0 win over the 10-man visitors.
David Moyes' team had Aaron Cresswell sent off for preventing a clear goalscoring chance early on, before Borre put Oliver Glasner's hosts in front with a neat finish after 26 minutes.
West Ham struggled to create clear-cut chances despite putting in a spirited performance, as their strong European run came to a disappointing end with a 3-1 aggregate defeat.
Despite sitting 11th in the Bundesliga, Frankfurt could end the campaign by securing a major European trophy and a spot in next season's Champions League by winning the May 18 final in Seville.
After a scrappy start, West Ham were dealt a huge blow when Cresswell hauled Jens Hauge down on the edge of the area 18 minutes in, with the VAR advising referee Jesus Manzano to send off the left-back before Filip Kostic drove the resulting free-kick wide.
The hosts needed less than 10 minutes to make their numerical advantage count, as Borre swept a side-footed finish into the bottom-left corner after meeting Ansgar Knauff's cut-back.
The Hammers struggled to assert themselves in a boisterous atmosphere, but almost found a surprise equaliser when Evan Ndicka cleared off the line after Jarrod Bowen's free-kick struck Kurt Zouma at the back post.
Borre hit a left-footed volley into Alphonse Areola's arms immediately after the break, before Craig Dawson headed Michail Antonio's cross straight at Kevin Trapp after an hour.
Moyes was sent off for a touchline outburst as West Ham's European dream slipped away, before Tomas Soucek missed a glaring headed chance in the final minute as the hosts cruised into the final.
What does it mean? Frankfurt build on first-leg triumph for historic semi-final win
Frankfurt's victory saw them reach their first European final since the 1980 UEFA Cup, when they beat Borussia Monchengladbach on away goals after a two-legged 3-3 aggregate draw.
With the win, Glasner's men have also become the first German team to reach finals in both the Europa League and UEFA Cup.
Hammers left deflated after Cresswell red
West Ham's hopes of reaching their first major European final since the 1975-76 Cup Winners' Cup were dealt a monumental blow when Cresswell became the first English player to ever receive two red cards in a single season in a European competition.
Cresswell was also sent off against Lyon in the last round, and is the first English player to be shown red in a European semi-final since Chelsea's John Terry in 2012 (against Barcelona in the Champions League).
Borre continues continental run
As well as recording seven goals and four assists in the Bundesliga this season, Borre has been in inspired form in Frankfurt's Europa League knockout games.
His composed finish means he has either scored (two) or assisted (one) three of Frankfurt's last five Europa League goals.
What's next?
Frankfurt host Borussia Monchengladbach in the Bundesliga on Sunday, while West Ham must turn their attentions back to securing European football for next season when they travel to Norwich City in the Premier League on the same day.
Celtic 3-1 RB Leipzig: Kuhn double inspires Hoops to Champions League fightback
Celtic's Nicolas Kuhn scored twice as the Bhoys fought back to clinch an impressive 3-1 victory over RB Leipzig in the Champions League on Tuesday.
Coming off a goalless draw at Atalanta on matchday three, Celtic recorded their second home win in the Champions League this season, moving up to 13th in the standings on seven points.
They had to do things the hard way, falling behind to Christoph Baumgartner's close-range header after a corner came off Cameron Carter-Vickers in the 23rd minute.
However, Kuhn levelled in the 35th minute with a superb curling effort from the edge of the box, then tapped into an open net for his brace shortly before half-time.
Benjamin Sesko and Baumgartner went close to equalising after the break but Celtic's birthday boy, 38-year-old goalkeeper Kasper Schmeichel, pulled off a couple of fine saves before Reo Hatate sealed victory with a close-range finish in the 72nd minute.
Leipzig's wait for a first point in the Champions League this campaign continues, with their chances of reaching the knockout stages fading after four defeats in as many games.
Data Debrief: Celtic flying high
Celtic may have been humiliated in a 7-1 defeat to Borussia Dortmund on matchday two, but that was a mere blip in what has so far been an impressive European campaign.
Their total of seven points is already their most in a Champions League campaign since 2012-13, when they collected 10, while they last netted more goals (currently nine) in a Champions League/European Cup campaign in 1977-78 (13).
Borussia Dortmund 1-0 Sturm Graz: Malen late show extends Champions League home record
Donyell Malen netted an 85th minute winner to hand Borussia Dortmund a narrow 1-0 victory over winless Sturm Graz in the Champions League on Tuesday.
A dominant showing from Nuri Sahin's side made it three wins from four games so far in the competition, moving them up to fourth in the new 36-team league.
Dortmund started brightly at the Westfalenstadion but were unable to find a breakthrough, with Marcel Sabitzer, Maximilian Beier and Serhou Guirassy going close.
They recorded 13 shots in the first half, their most in the opening 45 minutes of a Champions League match without scoring since Opta recorded this data (since 2003-04).
But Sturm Graz offered a threat on the break after the interval, with Jusuf Gazibegovic curling an effort narrowly wide before Mika Biereth missed a header from close range.
However, Dortmund left it late to seal the win as Malen pounced on Emanuel Aiwu's loose pass before being sent through by Guirassy, with the Dutchman rattling the ball beyond
Kjell Scherpen to seal another Champions League home win.
Data Debrief: Home comforts
While Sahin has come under fire for Dortmund's displays away from home, there is no arguing with his record at the Westfalenstadion.
He became the first Dortmund coach since the Bundesliga was founded to win each of his first seven home matches in charge (all competitions), while BVB are now unbeaten in 13 consecutive Champions League home games (W9 D4) - an extended club record.
Malen’s winner was timed at 84:56, Dortmund’s latest winning goal in a Champions League match since Kevin Groskreutz netted against Marseille in December 2013.
Lille 1-1 Juventus: French side's resistance broken after second-half penalty
Lille held firm to hold Juventus to a 1-1 draw at home on Tuesday in the Champions League, extending their unbeaten run to nine matches.
Jonathan David had fired the French club ahead in the first half, before Dusan Vlahovic scored a second-half penalty to level things up and leave both sides on seven points after four matches.
Things didn't start well for the home side, with Teun Koopmeiners thinking he had scored the opener for Juventus only for it to be disallowed for offside.
David put Lille ahead just three minutes later with their first chance of note in the match. The Canadian turned home to convert from a slick counter-attacking move.
Goalkeeper Lucas Chevalier had to be on form to keep the home side ahead before half-time, sensationally saving a Vlahovic half-volley 10 minutes before the break. Koopmeiners, meanwhile, had another effort chalked off in the 42nd minute.
Juventus upped the pressure in the second half, with Chevalier also upping things and pulling off two impressive saves in quick succession.
But ultimately, the Lille resistance was broken by a Vlahovic penalty on the hour after Francisco Conceicao had been fouled by Benjamin Andre.
The result leaves Juventus 10th and Lille 12th in the Champions League table.
Data debrief: Lille looking good for qualification
After drawing 1-1, Lille are unbeaten in nine matches in all competitions and three matches in the Champions League.
The French side had beaten Atletico Madrid and Real Madrid in back-to-back European games.
They now have a seven-point buffer between themselves and the bottom eight in the league table.
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