Weatherby, who was selected in the fourth round (No. 102) of the 2018 NHL Draft, tied it 2-2 with a power-play goal at 13:17 of the second period, taking a pass from Erik Karlsson and shooting glove side on Connor Hellebuyck.
He is the first Sharks rookie to score a goal in his NHL debut since Timo Meier did it against the Montreal Canadiens on Dec. 16, 2016.
"Weatherby [had a] big game, I was happy for him," Sharks forward Tomas Hertl said.
Hertl also had a goal and an assist, and William Eklund had an assist in his NHL debut for San Jose, which was playing its season opener. Adin Hill made 20 saves.
"You couldn't ask for a much better way to start the season. It was a team win," Sharks coach Bob Boughner said. "We tried to play to our identity as best we could. I think everybody knows there were some nerves there at the beginning, especially for some of the young guys."