Hazard beat Kevin Trap from 12 yards to seal victory for the Blues after Kepa Arrizabalaga had saved from Goncalo Paciencia, sparking scenes of celebration at Stamford Bridge.
Frankfurt, in contrast, were left distraught at the agonising defeat following a brave performance across both legs.
Chelsea now go through to face Arsenal in an all-English final in Baku at the end of this month. With Liverpool and Tottenham reaching the Champions League final earlier this week, it is the first time that all four places in the two European finals have been filled by one country.
There was little sign of the drama to come earlier in the evening. Frankfurt had the first opportunity when top scorer Luka Jovic headed straight at Kepa from a corner.
Chelsea had appeals for a penalty turned down when Danny Da Costa challenged Olivier Giroud. Da Costa was soon in action at the other end when he met Jovic’s cross from the left with a sweetly struck volley that was superbly tipped over by Kepa.
David Luiz then saw his header cleared off the line by Makoto Hasebe after Willian’s deflected free-kick from the left.
A goal was coming at this stage, so it was no surprise when the Blues did break the deadlock shortly before the half hour.
Hazard was able to turn away from David Abraham on the left flank before playing in Ruben Loftus-Cheek in the area, with the England midfielder opening his body before sweeping the ball past Trap for his 10th goal of the season.
Frankfurt stunned Stamford Bridge at the start of the second half when Jovic coolly finished off a fine move from just inside the area for his 27th goal of the season to level the tie on the night and on aggregate.
Chelsea responded when Hazard set up Giroud who lashed wide before Willian saw his cross-shot well saved.
Cesar Azpilicueta was perhaps lucky to only escape with a yellow card for a studs-up tackle on Filip Kostic late on, as neither side could find a winner in 90 minutes.
Sebastien Haller went close twice for Frankfurt in extra-time but his efforts were cleared off the line by Luiz and Davide Zappacosta.
Azpilicueta thought he had scored a dramatic winner with three minutes remaining but the referee adjudged the Spaniard had fouled goalkeeper Trapp before bundling the ball over the line.
With neither side able to find a winner the match went to penalties. Azpilicueta and Martin Hinteregger both saw their efforts saved before Paciencia missed Frankfurt's second spot-kick, allowing Hazard to fire Chelsea into the final.