Eintracht took the lead through their top scorer Luka Jovic, who superbly steered in Filip Kostic’s ball.
David Luiz also went close when his dipping free-kick came off the bar.
The second leg is on Thursday, 9 May.
There was a great deal of surprise when the Chelsea team-sheet showed Hazard’s name among the substitutes.
Manager Maurizio Sarri told BT Sport that he rested his 19-goal forward because of fixture congestion, with the Belgian having played 10 matches in a row. So it was a penny for Hazard’s thoughts when he was shown watching his team struggle and a goal down after 23 minutes.
However, Sarri did not rush on his key attacker and instead kept faith with what he had on the pitch, and was repaid with an excellent display thereafter.
Ruben Loftus-Cheek, brought back into the XI, led the fightback. After Pedro went close with a strike that drifted past Kevin Trapp’s left-hand post, the England player kept the goalkeeper on his toes with a strike that also flirted with the woodwork.
The pair then combined for the goal. Eintracht’s defenders lost possession after dawdling on the ball in the area, and Loftus-Cheek carved out an opening for Pedro to do the rest.
The Blues midfielder went close twice more, first with an effort that swept over the bar and then with a stinging shot with the outside of his boot that tested Trapp.
Brazilian Luiz went even closer with an blockbuster of a free-kick that dipped wickedly and crashed off Trapp’s bar. Hazard, who came on in the 61st minute, then set up another opportunity for the defender, but this time he headed straight at the German keeper.
They should have won the game but will no doubt be content with how they began their ninth major European semi-final since the Roman Abramovich era began in 2003. A repeat display at Stamford Bridge should see them reach their sixth European final.
