Osimhen netted either side of half-time in another talismanic display for the runaway Serie A leader, which sealed a 5-0 aggregate triumph across a tie overshadowed by violent clashes between supporters.
Piotr Zielinski won and converted a penalty to make it 3-0 as Eintracht toiled without suspended forward Randal Kolo Muani at the Stadio Diego Armando Maradona.
After swatting last season's Europa League winners aside, Napoli will be seen as a team to avoid when they join fellow Italian giants Milan and Inter in Friday's last-eight draw.
Napoli made a typically fast start as Matteo Politano tested Kevin Trapp with a low drive in the first minute, before the goalkeeper denied Khvicha Kvaratskhelia at the end of slaloming run.
Oliver Glasner's well-drilled visitors kept Napoli at bay for much of the first half, though they were again indebted to Trapp for keeping out Kvaratskhelia's prodded attempt 40 minutes in.
Eintracht was unable to make it to half-time, however, with Osimhen hanging in the air to head Politano's cross into the top-right corner after a fine pass from Stanislav Lobotka.
Osimhen needed just eight minutes to grab his brace after the restart, stretching to divert Giovanni Di Lorenzo's low cross home from inside the six-yard box.
Trapp denied Kvaratskhelia for a third time when the Georgian attempted a curling shot on the hour mark, but he was unable to prevent Zielinski from making it 3-0 from 12 yards after the midfielder was clumsily tripped by Djibril Sow.