Meier scored on a long shot from the left side inside the blue line that went off the far post and in to make it 2-0 at 13:44.
Hertl extended it to 3-0 at 17:24. Alexander Barabanov won a race for the puck in the neutral zone and passed to a trailing Hertl, who scored from the slot.
Skinner cut it to 3-1 at 7:56 of the second period, picking up the puck along the right wall and skating to the high slot before scoring top corner on Hill.
"I just think we need more urgency, and that's not really rust, maybe," Skinner said. "We were sort of waiting for plays to develop and as a result we were kind of on our heels for a bit there."
The Sabres pushed in the third period, outshooting the Sharks 22-5 and drawing two penalties.
Girgensons made it 3-2 on the power play at 18:14 of the third period, jamming in a rebound at the left side of the net with Luukkonen pulled for the extra attacker.
"It's tough to take confidence back when you give it up," Granato said. "We elevated. We competed harder. I really, really pushed our guys at the end of the period. I was disappointed. I didn't think we were competing hard enough. We had some individual errors, but I wanted to make sure we didn't dwell on it. … We played with too much fear in the first period, which is hesitation. We needed to get rid of that. We did a better job the second two. All the frustration you hear in my voice is that."
NOTES: Hertl has 15 points (nine goals, six assists) in his past 11 games. … Meier has a goal and four assists in a three-game point streak. He has three goals and five assists in his past six games. … Skinner has 11 points (nine goals, two assists) in his past 11 games against the Sharks. … Forward Ryan MacInnis played 10:00 and had two shots on goal in his Sabres debut.