Shea Theodore, Max Pacioretty and Alex Tuch also scored for Vegas, which had not beaten St. Louis in regulation in their previous 11 meetings dating to 2017.
''We had a lot of things stacked against us,'' Vegas coach Pete DeBoer said. ''Six games in 10 nights, and all the stuff we dealt with on the road from being kicked out of our hotel to false positive tests. To end up coming out the end of this trip 4-2 in some tough buildings is a great road trip.''
Fleury made his 15th start in Vegas' last 16 games.
''It's not easy for goalies to play back-to-back,'' Stone said. ''He came in and had his best game of the trip. That just shows the leadership and quality of goalie it is. I think it lifts our bench.''
Ville Husso made 21 saves for the Blues and took his first regulation loss since Feb. 6 against Arizona. He is 0-2 on home ice.
''They're a good team over there but we feel that we can play with them,'' Blues center Brayden Schenn said. ''We can compete with them, and I don't feel we showed that through these two games playing against them.''
St. Louis is 4-7-3 at home, compared to 10-2-2 away from Enterprise Center.
''We've got to be able to produce at home,'' Blues coach Craig Berube said. ''We've got to find a way to execute here at home and score goals on a consistent basis.''
Theodore scored his fourth goal of the season, and first since Jan. 22, on the first shot of the game, giving Vegas a lead 4:01 into the first period.
Stone followed with his eighth goal of the season 1:32 into the third period.
Sammy Blais ended Fleury's bid for his 66th career shutout when he scored his third goal of the season on a feed from Vladimir Tarasenko. Tarasenko has a goal and three assists in four games since returning from shoulder surgery.
Pacioretty gave Vegas a 3-1 lead on a pass from Chandler Stephenson 7:25 into the third period.
St. Louis failed to convert on any of its four power plays, including a 53-second two-man advantage midway through the third period. The Blues entered the game having scored with the man-advantage in six straight contests.
''For us it was about disrupting them,'' Stone said. ''I think when we box out and block shots and (Fleury) can see the puck it just makes his life easier.''
Stone added his second goal of the game on a wraparound with 7:51 remaining, stretching the Vegas lead to 4-1.
Tuch scored into an empty net with 30 seconds remaining for his 13th of the season.
WINNING THE BATTLES
Vegas has not lost when holding the lead at the end of a period. The Golden Knights are now 10-0-0 when leading after the first period and 12-0-0 when leading after two periods.
Blues center Dakota Joshua participated in his first NHL fight when he dropped the gloves with Keegan Kolesar in the first period. Kolesar took offense to Joshua's big hit on Ryan Reaves behind the Golden Knights net. Joshua and Kolesar each received five-minute fighting majors at the 8:32 mark.
Vegas: Home for the first of two games against San Jose on Monday night.
St. Louis: At Los Angeles on Monday night in the opener of a six-game trip.
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