Later in the period, Golden Knights forward Jonathan Marchessault scored to make it 2-1 at 13:09 on another one-timer on the power play after Tomas Hertl was called for roughing at 12:47.
Reilly Smith scored 1:17 later to give the Golden Knights a 3-1 lead heading into the third period.
"They feel good about their game when they're making plays and that's when we took some undisciplined penalties," Sharks forward Logan Couture said.
San Jose had few answers for the Golden Knights top line of Smith, William Karlsson (who scored the game-winning goal at 8:17 in overtime) and Marchessault, which combined for seven points (three goals, four assists) in Game 3.
"They're quality players," Dillon said. "But I think for us, we've got a game plan, but I don't think we've completely executed it yet. We're kind of doing it in bits and pieces."
Sharks coach Peter DeBoer did not think the penalties were the only issue Monday, pointing out that they had dominant stretches in the first period but couldn't score, something they will need to fix ahead of Game 4.
"To come out of that first period 0-0, we need to find a way to get the lead," DeBoer said. "We're chasing the game the whole series here. Even when we have a quick start like tonight, [Golden Knights goaltender Marc-Andre Fleury] is shutting the door and finding the way to let them hang around until they can get something going, which is what happened.
"They got a power play in the second and got rolling a little bit. I think our game is in a good place. Just one of those games we're doing a lot of good stuff, but through the three games we're chasing the game every night. We've got to find a way to get out in front."