McElhinney, who signed a two-year contract with Tampa Bay on July 1, has 11 shutouts in the NHL. It was his first shutout since Feb. 23, 2019 for the Carolina Hurricanes.
"It's great," McElhinney said. "Obviously it's been kind of up and down for me personally, but it was nice to get in there and solidify this trip with two points."
WATCH: [All Lightning vs. Sharks highlights]
Steven Stamkos had a goal and an assist for the Lightning (32-15-5), who are 15-2-1 in the past 18 games.
Aaron Dell made 32 saves for the Sharks (22-27-4), who have lost five of six.
"We played a hard game, and I'd like to see us get off to a better start," Sharks coach Bob Boughner said. "But the penalty kill gave us a chance. Dell gave us a chance, he was great all night. We just couldn't find that goal."
Stamkos made it 1-0 with his 23rd goal of the season at 4:04 of the second period on a shot from the left circle off a cross-ice pass from Kevin Shattenkirk. Stamkos has 12 points (six goals, six assists) in a seven-game point streak.