Janis Moser had a goal and two assists, and Andrei Vasilevskiy made 22 saves for the Lightning (3-0-0).
Jack Eichel and Shea Theodore each had two assists for Vegas (3-2-0). Hill made 21 saves.
"I think we have an idea of the areas we need to be better on," Golden Knights coach Bruce Cassidy said. "We've protected leads well, so hopefully this is a one-off, but this happened last year in this building. It's happened to us more on the road than it should."
Brayden McNabb gave Vegas a 1-0 lead at 5:26 of the first period with a shot from the high slot that beat Vasilevskiy glove side.
"We should have taken that to overtime," McNabb said. "It's too bad. We need to do better in those situations. We have to finish games and have killer instinct. We're playing the right way ... but every little detail kind of matters."
Moser tied it 1-1 at 10:21, taking the puck against Brett Howden and beating Hill five-hole. It was the defenseman’s first goal with the Lightning since being acquired from the Utah Hockey Club on June 29 along with forward Conor Geekie and two draft picks for defenseman Mikhail Sergachev.
"First goal as a Bolt is a wonderful feeling, first goal with any new team is nice," Moser said. "Obviously, it has its challenges to get adapted to a new team, new system, new teammates, but I feel like it's going pretty well so far."
Pavel Dorofeyev put Vegas ahead 2-1 at 15:20 with a power-play goal, a one-timer from the right circle off a pass from Theodore.
Kucherov tied it 2-2 at 19:47 when he finished off a 2-on-1 with Jake Guentzel.
"First of all it was two very good hockey teams playing a game, but I thought we started to push in the second period and we pushed in the third," Cooper said. "But the big difference between tonight and our previous two games is we really managed the puck."