The Nigerian international becomes the foreigner with the most European competition goals in the Turkish side’s history with 13 goals.
Czech Republic legend Milan Baros had held the record for more than a decade with 12 goals during his spell at the Rams Stadium.
Trailing from the first leg in Turkey, Juventus entered the return fixture knowing they needed three goals to overturn the deficit and started with urgency and intent.
Their bright opening was rewarded in the 37th minute when captain Manuel Locatelli calmly converted from the penalty spot after Khephren Thuram was fouled by Lucas Torreira.
However, the Italian giants suffered a major blow just four minutes into the second half as Lloyd Kelly was shown a red card for a reckless challenge, reducing them to ten men.
Despite the setback, the Serie A side fought back bravely, with Federico Gatti and Weston McKennie scoring to level the tie and force extra time.
Osimhen delivered the defining moment to help Okan Buruk’s men qualify for the next round. The former Napoli striker found space inside the area and rifled a low shot through the legs of Mattia Perin with Baris Alper Yılmaz adding another.
The former African Player of the Year, who now has seven goals in this season’s campaign, capped a commanding display that included five shots and 15 passes at 66.7 percent accuracy, earning him the man-of-the-match award.
"This match was disappointing from the start. Congratulations to Juventus, they really pushed us hard. We knew this was a tough pitch," the Nigerian told the media.
"Even with 10 men, they pushed hard and scored two goals. I'm happy with my goal and what I did. We know we have a lot to work on.
"Juventus deserved to win, but we deserved to advance to the next round. Even in the most difficult moments, the fans were with us.
"Galatasaray fans deserve this happiness to the fullest. Even when we were behind, they never stopped supporting us.
"We can say that we advanced to the next round thanks to them. We will continue to work. We need their support very much. We have a long way to go, but we are sure that they will be with us every step of the way."
Galatasaray will face either Tottenham Hotspur or Liverpool in the Round of 16 when UEFA holds the draw on Friday.
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