Both sides squandered chances to win it as a frenetic first leg ended all square in Frankfurt.
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Chelsea extended their unbeaten UEFA Europa League run to a record 16 in Frankfurt but were perhaps left wanting more after failing to press home their advantage.
Maurizio Sarri's team were on top for much of the evening in Germany, though fell behind during the hosts' best spell of the game. Luka Jović – who else? – did the damage, diving to guide a header inside the post.
Pedro Rodríguez restored parity just before the break, and Chelsea spent much of the second period camped inside the Frankfurt half but could not break through. David Luiz's free-kick against the bar was as close as they came.
View from the stadium: James Thorogood
An electric atmosphere and a fiercely-fought encounter made for a captivating first leg. Frankfurt's high-octane style caught Chelsea by surprise and Jović duly got his goal. However, the hosts had little answer to the relentless pressure they faced after taking the lead and can be pleased to have escaped with just one away goal conceded.
Man of the match: Ruben Loftus-Cheek (Chelsea)
From his inventive, eye-catching first touches, it was clear that Loftus-Cheek was in the mood to stamp his authority on one of the biggest games of his club career, this powerful performance in midfield showcasing his rare marriage of gifts.
When at his best like this, the 23-year-old is as light on his feet as Eden Hazard yet as direct as Olivier Giroud, never more evident than when his svelte touch following fine hold-up play created Pedro's equaliser. The England international then twice came close to scoring a Chelsea winner.
16: Chelsea extended their unbeaten UEFA Europa League run to a record 16 games, surpassing Atlético Madrid's mark set in 2011 and 2012.
9: Jović took his tally to nine in the UEFA Europa League this season, second only to Olivier Giroud.
12: Frankfurt have never lost at home in the UEFA Europa League (W9 D3).
13: Chelsea are 13 games unbeaten in this UEFA Europa League campaign (W11 D2).
8: The Blues are eight unbeaten away from home in the UEFA Europa League.
39: Frankfurt's fourth major European semi-final came 39 years after their third; Chelsea managed 11 in the meantime.
31: Chelsea have scored a competition-leading 31 goals this season; Frankfurt are second on that list with 29.
9: Chelsea have won their last nine UEFA Europa League home matches.
Why it's all still to play for
Chelsea are formidable at home in Europe – the Blues have never lost at Stamford Bridge in this competition, winning their last nine – yet they were almost shocked by Slavia Praha in the last eight, holding on for a nervy victory.
Frankfurt will be buoyed by the return of Ante Rebić in London, while their results at Shakhtar Donetsk and Internazionale in the last 32 and last 16 respectively would take them to the Baku showpiece.
It is advantage Chelsea but it is far from beyond this spirited Eintracht side to pull off a heist – there have been plenty of eyebrow-raising second-leg triumphs already in Europe this season, after all.
There's no respite for Frankfurt as they travel to Leverkusen on Sunday evening. Hütter's side are fourth in the Bundesliga, three points clear of both their opponents and Mönchengladbach with three to play. The top three are decided.
Chelsea are at home to Watford on Sunday afternoon and can effectively wrap up a top-four spot if they win and Arsenal, in action against Brighton, don't.