Jonathan Bernier made 23 saves for Detroit (16-24-6), which had a season-high three-game winning streak end. The Red Wings won 4-1 here Thursday to open the two-game set.
"In the first game of the series, they gave us some chances and we took advantage of them," Red Wings captain Dylan Larkin said. "Tonight, we were the ones making key mistakes."
The Blackhawks are two points behind the Nashville Predators for fourth place in the eight-team Discover Central Division. The top four teams will qualify for the Stanley Cup Playoffs.
Chicago plays Nashville in its next three games beginning Tuesday.
"Obviously, we all know where we are in the standings," Hinostroza said. "To win a game like this, 4-0 where we battled the whole game, that's great going into this series we've got with Nashville.
"These are the three biggest games of the year."
Kane gave the Blackhawks a 1-0 lead with 21 seconds left in the first period, scoring his second goal in 11 games. Hinostroza forced a turnover by Troy Stecher in the Red Wings end before setting up Kane.
"[Kane] is such a smart player that it is an honor to play with him," Hinostroza said. "He's so good with the puck that you are always working hard to use your speed and get open to make things happen."
Alex DeBrincat made it 2-0 at 12:40 of the second period, knocking in a loose puck after Connor Murphy's point shot hit teammate Brandon Hagel in front of the net. DeBrincat, who grew up in suburban Detroit, has scored 20 points (eight goals, 12 assists) in 14 games against the Red Wings.
Wyatt Kalynuk made it 3-0 at 1:16 of the third period off a pass from Hinostroza. Pius Suter scored into an empty net at 15:27 for the 4-0 final.
