The Nigerian international maintained his red-hot form for the Istanbul giants, sending Jose Mourinho’s side crashing out of the competition.
Osimhen fired Galatasaray into the lead at the Ulker Stadium Sukru Saracoglu just ten minutes in, latching onto a setup from Barıs Yılmaz.
The former African Player of the Year doubled his tally from the penalty spot in the 27th minute. In a thrilling first half, Sebastian Szymanski pulled one back for Fenerbahce just before the break.
The fiercely contested clash produced three red cards, but that had little impact on the game's flow.
Man of the Match Osimhen was substituted for Alvaro Morata with four minutes remaining.
Galatasaray will now take on Konyaspor in the semi-finals, scheduled between April 22 and 24.