Detroit (26-30-11) has lost five in a row (0-4-1) and trails the Columbus Blue Jackets by 12 points for the second wild card into the Stanley Cup Playoffs from the Eastern Conference. Jimmy Howard made 28 saves.
"We weren't good enough," Red Wings coach Jeff Blashill said. "There have been a lot of games this season where we have competed hard and not gotten the results we deserve. Tonight, our compete level was unacceptable."
Tuch scored unassisted to give the Golden Knights a 1-0 lead at 2:24 of the first period, keeping it on a 2-on-1 rush and beating Howard to the short side.
Vegas scored three times in the second period.
Eakin scored shorthanded to make it 2-0 at 10:40, taking advantage of a bad Detroit line change to score on a breakaway.
"We had a turnover that should have never happened, and then tried to make a line change," Blashill said. "That's discouraging to see."
Eakin scored his second goal of the period at 17:48 to put the Golden Knights ahead 3-0, circling through the Red Wings zone before beating Howard with a wrist shot.
Tuch made it 4-0 when he scored his second goal of the game at 18:38.