The Blues wasted numerous opportunities to net more away goals, although they were perhaps fortunate to avoid suffering a late defeat after Frankfurt rallied during the closing minutes.
Chelsea started brightly in Germany and looked to take the game to their hosts, but Frankfurt quickly began to show that they were not intimidated by featuring in their first European semi-final for 39 years.
David Abraham should have done a lot better with a volley from 15 yards out after receiving the ball from a set piece, but Frankfurt deservedly took the lead at the midway point of the first half.
Filip Kostic's cross from the left found Jovic by the penalty spot, and the in-demand striker managed to direct a brilliant glancing header into the far corner of the net from 12 yards out.
Chelsea improved after going behind with Pedro sending a 25-yard volley marginally wide of the post, but Frankfurt were still making it difficult for the visitors in the final third.
That changed as the game approached half time with Ruben Loftus-Cheek having a bigger impact on the match, and the England international went close to getting on a scoresheet with a low strike from the edge of the area which went inches wide of Kevin Trapp's right-hand post.
Loftus-Cheek also provided the assist for Chelsea's leveller just before the break. The midfielder kept the ball alive after a set piece before teeing up Pedro, who took a touch and then sent a 15-yard strike into the bottom corner of the net.
Chelsea went straight back onto the front foot after the restart, and the visitors squandered an opportunity to go in front when Loftus-Cheek hooked a first-time effort off target from 12 yards.
Loftus-Cheek saw a half-volley comfortably palmed away by Trapp, while Luiz somehow only found the hands of the goalkeeper after sending a header towards the far corner from Hazard's set piece.
Many of Frankfurt's players looked exhausted as the game entered the closing stages, but Adi Hutter's side found a second wind and should have regained their advantage.
A corner from the right was perfect for Abraham to power a header into the net from eight yards out, but the defender somehow failed to hit the target under little pressure from the Chelsea backline.
The visitors also appeared to fade as the game reached a conclusion, but Sarri and his squad will be relatively content that they can head into the reverse fixture at Stamford Bridge with a slender advantage.
EINTRACHT FRANKFURT (3-4-1-2): Trapp; Abraham, Hasebe, Hinteregger; Da Costa, Rode, Fernandes (Paciencia 73'), Falette; Gacinovic (Willems 92'); Jovic, Kostic
CHELSEA (4-3-3): Arrizabalaga; Azpilicueta, Christensen, Luiz, Palmieri; Kante, Jorginho, Loftus-Cheek (Kovacic 83'); Willian (Hazard 61'), Giroud, Pedro