Hiller was outstanding in the second game of back-to-back starts, earning his third shutout of the year and the 19th of his career. 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.
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 just one playoff round since earning its only Stanley Cup title in 2007.
The Ducks were just the seventh team in NHL history to win 16 games in a 17-game stretch, and their latest victory is their seventh straight since a loss to San Jose on Dec. 29. Anaheim had scored at least four goals in each of its last six games, but didn't need more than Cogliano's score to hold off the Red Wings.
Hiller's winning streak is the NHL's longest in two decades, and trails only Gilles Gilbert's 17-game run for Boston during the 1975-76 season in NHL history.
He 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.
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.
"Being self-serving, I don't want him hurt very long," Babcock said Saturday night. "But it'd be nice if he missed tomorrow."
Even after grinding out a win over the Kings on Saturday night at the same time Anaheim coasted past Phoenix, Detroit had more energy during a scoreless first period and again in the second, but Hiller was unbeatable.
The Ducks went ahead after a neutral-zone turnover ended up with Saku Koivu, who got the puck to Cogliano in front for his 15th goal. The speedy Cogliano, who got a four-year contract extension recently, is 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.
NOTES: 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. ... 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. ... Anaheim played its fourth straight game without D Bryan Allen, who has a lower-body injury.