William Ecklund had two goals, and Luke Kunin and Tyler Toffoli also scored for San Jose. who have won two straight and are 3-1-2 in their last six games. Mackenzie Blackwood finished with 16 saves.
Dylan Larkin had a goal and had an assist, and Alex DeBrincat, Marco Kasper and Michael Rasmussen also scored for Detroit, which has lost five of six. Cam Talbot stopped 29 shots.
Detroit trailed 4-3 with 2:06 left in regulation when DeBrincat tied it on a power-play goal.
Celebrini, the 2024 No. 1 overall draft pick, then won it in the extra period with his fourth goal of the season and first career overtime score.
Red Wings: Detroit led 3-2 midway through the second period when Michael Rasmussen was credited with the score on an own goal, but they were unable to close the game.
Sharks: The rebuilding Sharks are 6-5 since losing nine straight to start the season. They trailed most of the game but played inspired hockey, especially late, as they rallied for the win.
DeBrincat sent the game to overtime on a power-play goal that came 1:40 after Ty Dellandrea was assessed a two-minute penalty for tripping Erik Gustafsson.
The Sharks outshot the Red Wings 34-20.
Red Wingst host the New York Islanders on Thursday, and Sharks visit Dallas on Wednesday.