Then the Red Wings broke through after getting a break and making one of their own to force overtime.
Hiller had a chance to extend the game even deeper into Monday night, but he guessed wrong and it was costly for the Anaheim Ducks.
He stayed in net -- instead of leaving it to play the puck sooner -- and Damien Brunner capped a blur of a sequence with a goal 4:50 into overtime. That lifted Detroit to a 3-2 series evening win over Anaheim in Game 4.
"The longer you can put this off, the better I like our chances," Red Wings coach Mike Babcock said.
Detroit defenseman Jakub Kindl started the winning play, passing the puck from behind Detroit's net up the ice to Joakim Andersson. He tried to connect with teammate Gustav Nyquist only to have Hiller poke the puck away.
Brunner, playing in his first NHL season, was in perfect position in front of the net and swatted in the puck to end the game.
"It's a tough call for the goalie," Ducks coach Bruce Boudreau said, defending his goaltender's decision. "He has to anticipate who is going to win the race. It was a 50-50 shot, so it's better to stay in the net.
"He made the saves, but the other guys on the ice didn't pick up Brunner coming in."
Game 5 is Wednesday night in Anaheim and Game 6 is back in the Motor City on Friday night. During the Red Wings' run of playing in 22 straight postseasons, they have won 11 of 19 series that were tied at 2-2.
"We missed a great opportunity," Boudreau said. "We got the lead twice in the game and we didn't hold it."
Hiller, who shut out Detroit in Game 3, finished with 46 saves. It was his second-highest total in the playoffs, trailing only the 59-save performance he had in a triple-OT win against Detroit in 2009.
The Swiss goalie wished he made at least one more save, saying he tried to clear the puck when it appeared to get away from Nyquist.
"It's definitely tough to lose that way," Hiller said.
Jimmy Howard had 31 saves for the Red Wings.
Detroit played with a sense of urgency, hoping not to face elimination on the road in Game 5. The Ducks, meanwhile, didn't make their coach happy with their effort.
"They played very good, very hard, very desperate," Boudreau said of the Wings. "I don't think, quite frankly, we had 20 players playing. We had some passengers."
Beleskey came to play. He hit Detroit defenseman Kyle Quincey hard on the left side of the net and had enough space to shoot and score off his own rebound to put the Ducks ahead 1-0.
The Red Wings were outshooting Anaheim 27-15 through two periods, but had nothing to show for it.
Smith changed that thanks to some good fortune, ending a 101:18 scoring drought that started after Detroit's 5-4 overtime win in Game 2. Smith's shot from just inside the blue line went off Ducks defenseman Francois Beauchemin and past Hiller 1:18 into the third period.
Steckel netted a go-ahead goal midway in the third period.
The Red Wings, and their fans, thought Johan Franzen scored moments later, but referee Eric Furlatt waved it off and his call was upheld by video review.
Datsyuk gave the red-clad, white-towel waving crowd a reason to cheer when he tied it. He carried the puck up the left side, made a move to the outside that created space between him and defenseman Luca Sbisa, and flicked a wrist shot past Hiller's glove that knocked the water bottle off the top of the net.
"You have to give them credit where credit's due, Datsyuk's a world-class player," Ducks defenseman Cam Fowler said. "He pretty much puts the team on his shoulders."
That was Datsyuks's first goal of the series after leading the team with 15 scores and 49 points during the regular season.
Ryan Getzlaf, who had two goals and four points in the first three games, had a chance to put the Ducks up after Datsyuk's score, but his breakaway backhander was stopped by Howard.
"Even though they outshot us, we had some Grade A chances that have to be successful if you're going to win," Boudreau said.
Detroit's Justin Abdelkader served his first of a two-game suspension for hitting Toni Lydman's head with his left shoulder. ... Lydman was scratched because of headaches and a sore neck and was replaced in the lineup by Sheldon Souray.