Eden Hazard stroked home the deciding spot kick after 90 minutes and extra time could not separate the two sides at 2-2 on aggregate.
Despite persistent criticism from his own fans the Italian has now taken Chelsea to two major finals in his first season and already secured qualification for the Champions League next year.
And he indicated after the dramatic events at Stamford Bridge that he has no intention of leaving the club.
When asked if a return to Italy was on the cards, he replied:
‘At the moment, no.’
Chelsea scored first on Thursday night through Ruben Loftus-Cheek to take a 2-1 lead on aggregate before Luka Jovic equalised early in the second-half.
Sarri admitted his players tired as the game wore on.
‘It was obvious that if we couldn’t finish it off in the first half, it’d be really tough, as Eintracht are a very dynamic side who exploit spaces well,’ Sarri told Sky Sport Italia.
‘We ran out of fuel in the second half, did well in extra time to hold out.
‘I thought it must’ve been very entertaining for the neutral, but it was a lot of suffering for us on the touchline.’