Howden rifled in the winner on a rebound from behind the goal, the puck appearing to ricochet into the net off Dallas goaltender Jake Oettinger.
Earlier, Jason Roberton had fired Dallas into the lead in the first period, cleverly tipping in Roope Hintz's feed to make it 1-0.
William Karlsson levelled for Vegas in the second period and then scored again at the start of the third to put the Knights up 2-1.
Hintz equalized for Dallas after latching on to a feed from Joe Pavelski and blasting a shot in off the post.
But Teddy Blueger restored the Knights' advantage with his first goal of the play-offs, bundling home from close range after a goalmouth scramble.
Jamie Benn hit back for Dallas, however, with 1:59 remaining in regulation to make it 3-3 and send the game into overtime before Howden settled it for the home side.
Game 2 in the series takes place in Las Vegas on Sunday.
