Hiller made 22 saves in his 14th consecutive victory, and the steamrolling Ducks beat the Detroit Red Wings 1-0 on Sunday night for their 17th victory in 18 games.
All they needed was a second-period goal by Cogliano and 60 outstanding minutes from Hiller, who earned his third shutout of the year and the 19th of his career.
"The win, that's what counts," Hiller said. "The other things are just something for yourself. ... It's nice to make a name for myself, but winning as a team is what it's about."
The Swiss Olympic team selection hasn't lost since Dec. 3, matching four other goalies for the second-longest winning streak in a single season in NHL history. His streak is the league's longest in two decades, and trails only Gilles Gilbert's 17-game run for Boston during the 1975-76 season.
Hiller hadn't posted a shutout during the Ducks' streak, and the Red Wings made him earn it all the way down to the frantic final minutes.
"Our goalies have been amazing for us lately, especially Jonas," said Saku Koivu, whose pass set up Cogliano's winning goal. "He made some key saves. The way he's been playing, he's give us some confidence. We know even if we're not at our sharpest, we're going to have a chance to win."
The Ducks are only the fifth team in NHL history to win 17 games in an 18-game stretch, and their latest victory is their seventh straight since a loss at San Jose on Dec. 29. Anaheim's powerful offense had scored at least four goals in each of its past six games, but didn't need more than Cogliano's score to hold off the Red Wings.
Petr Mrazek stopped 22 shots for the Red Wings in a rematch of last season's first-round matchup in the Western Conference playoffs. Detroit won Game 7 at Honda Center last spring to eliminate second-seeded Anaheim, which has won only one playoff round since earning its only Stanley Cup title in 2007.
Hiller was nearly matched by Mrazek, who got his first start since Dec. 23 after Jimmy Howard injured his hip while making 44 saves in Detroit's win over Los Angeles on Saturday night.
"We played good enough to win, but we couldn't find a way," Detroit captain Henrik Zetterberg said. "Mrazek played unbelievable in the net for us, and it was good to see him come in and play with that confidence. We got some opportunities on the power play, but couldn't score. Injuries happen, but we can't use that as an excuse. You have to deal with it."
Getzlaf sat out for the Ducks after blocking a shot with his right foot during Saturday's win at Phoenix. X-rays on the captain's foot were negative, and the absence of the NHL's fourth-leading scorer created mixed feelings for Detroit coach Mike Babcock, who will be Getzlaf's coach for Canada at the Sochi Olympics next month.
Detroit dropped to 2-2-0 on its five-game road trip, winning one of three in California.
"We've really competed hard in three of the first four games on the trip," Babcock said. "The reality is we've got to find a way to score. Our power play has got to be more dangerous."
Detroit had more energy during a scoreless first period and again in the second, but Hiller was unbeatable. The Ducks went ahead after a Detroit turnover ended up with Koivu, who got the puck to Cogliano in front for his 15th goal, leaving him just three goals shy of his career high.
Justin Abdelkader rang a shot off Hiller's post with less than five minutes to play, but Hiller comfortably survived a Detroit power play in the final minutes and earned his 23rd victory.
Detroit played without veterans Daniel Alfredsson and Pavel Datsyuk. Alfredsson has missed two games with back spasms, while Datsyuk has missed four straight with a lower-body injury. ... Anaheim played its fourth straight game without D Bryan Allen, who has a lower-body injury. ... The Ducks stuck with Hiller instead of giving an NHL debut to John Gibson, the touted 20-year-old goalie prospect recently recalled from the AHL. Gibson was brought up from Norfolk to back up Hiller so Frederik Andersen could get playing time in the minors.