The match, played at Rams Park, saw the hosts take an early lead in the seventh minute when Kayserispor defender Aaron Opoku scored an own goal.
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Osimhen then converted a penalty in the 25th minute to make it 2-0, marking his 200th career goal (165 for club and 35 for Nigeria).
He continued to influence the game by setting up Gabriel Sara for Galatasaray’s third in the second half.
The former African Player of the Year who has now scored 14 goals in 20 appearances this season, was substituted in the 82nd minute.
Mauro Icardi added a fourth from the penalty spot before full-time as the comprehensive victory keeps Galatasaray atop the Süper Lig with 49 points, six clear of second-placed Fenerbahce.