Pedro's first-half equaliser gave Chelsea a potentially crucial away goal after a hard-fought clash at the Commerzbank Arena.
Sarri raised eyebrows with his decision to leave Hazard out of the starting line-up, but he said he had no choice but to rest the Belgian star.
"Eden was on the bench because he has played 10 games in a row," said Sarri.
"It is impossible for a player to play 70 to 75 matches in a season. Sometimes they need to rest, or at least not to play for 90 minutes."
"We want to reach the final, and this time I want to win it," added Sarri, whose team lost the League Cup final against Manchester City earlier this season.
Sarri warns Chelsea of Eintracht threat
"Three months ago, we were in trouble. Now we are in the top four and in the semi-final, so I think we deserve to win a trophy."
Sarri said his team had made a "mistake" in the opening 20 minutes as they struggled to deal with a determined Frankfurt in the first half.
Fired on by the home crowd, the hosts caused problems for Chelsea early on before being outclassed for long periods in the second half.
"We started to play after the goal and dominated the game for 70 minutes," said Sarri.
Filip Kostic found the breakthrough for Frankfurt on 23 minutes, sending a pinpoint low cross to Luka Jovic, who ducked low to flick his header inside the far post.
Sarri urges Chelsea to finish on a high note
Frankfurt then tried to grab control of the tie, with Sebastian Rode snapping at Chelsea's heels in midfield and Jovic and Mijat Gacinovic lurking in wait for a chance to break.
Yet Sarri's side eased their way patiently back into the game, and deserved to draw level when Pedro finally found the net with a well-placed strike just before half-time.
"It was annoying to concede just before half-time, but they have quality in front of goal," said Frankfurt midfielder Gelson Fernandes.
It remained a fiercely contested game after the break, and Chelsea midfielder Jorginho had to be bandaged up after a clash of heads with Rode.
Always a step ahead of Frankfurt in the second half, the visitors nearly took the lead when David Luiz struck the bar with a fizzing free-kick.
Frankfurt had a penalty appeal waved away 12 minutes from time, and both David Abraham and substitute Goncalo Paciencia came close to snatching a late winner.
Sarri warns Chelsea of Eintracht threat
"I am proud. I don't know how many goals Eintracht Frankfurt would have conceded against such an opponent a few years ago," said Fernandes.
Frankfurt coach Adi Huetter, meanwhile, said that his team would have to be brave when they travel to Stamford Bridge for the return leg next Thursday.
"Chelsea will obviously be clear favourites, but we are optimistic that we can get something there -- everything is possible."
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