Samuel Pahlsson's goal with 14:06 remaining proved to be the difference for Anaheim, which meets Ottawa in the Stanley Cup Finals.
The Ducks and Senators, both looking for their first NHL championship, open Monday night in Anaheim.
Giguere finished with 26 saves. Hasek had 25.
The Ducks' goalie afterward expressed his respect for the 42-year-old Hasek, "doing what he does at his age."
"You know it's not easy. I won't be there when I'm his age, for sure. But I really see it as the Ducks beating the Red Wings. It's not about the goalies," Giguere said. "In fact there were a lot of goals (15 given up by each goalie) in this series.
"We have almost 25 guys on our lineup. So you need all these guys to win. And it's always the better group that's going to get out of it, not the better goalie, and I truly believe in that."
Hasek, a six-time winner of the Vezina Trophy as the NHL's best goalie during the regular season, gave Giguere more credit than the Anaheim goalie gave himself.
"He (Giguere) played great. I hate to say he played better than me, but maybe he did," Hasek said. "He stole Game 5, and that was the key to the series.
"You have to be a little bit lucky, and he was. He won the series for that team."
Hasek said he would decide in a few weeks whether he'll be back next season.
Giguere had made 36 saves in each of the previous two games. He kept the Ducks in Game 5 until Scott Niedermayer tied it with 48 seconds left and Teemu Selanne scored the game-winner in the 2-1 overtime victory.
The Red Wings outshot the Ducks 16-3 in the third period this time and got one goal from Henrik Zetterberg and two -- both on power plays -- from Pavel Datsyuk, the last with 3:04 remaining.
"They took the attitude that this was desperation and they started to control the puck," Anaheim coach Randy Carlyle said. "Our counteraction was that we didn't skate. We stood still, and we just kept feeding them the puck.
"It's kind of like a paralysis that takes place over your team."
Detroit coach Mike Babcock, who was the Ducks' coach when they played in the Cup finals in 2003, gave them credit.
"Games 4 and 5 where we played at every opportunity to win and let them slip away, that in the end cost us. You have to tip your hat to Anaheim," Babcock said.
"They've got a good team and they found a way to get it done, and they are going to play a good Ottawa team."
Anaheim star Teemu Selanne, who at 36 will be playing in his first Stanley Cup Finals, was afraid it was all slipping away when the Red Wings stormed back.
"In the third period, I have never been so nervous and so shaky in a hockey game in my life -- and I was on the bench," Selanne said.
"We did not want to go back to Detroit. When it ended, I took several deep breaths."
The Ducks lost Game 7 of the Stanley Cup Finals to New Jersey in 2003. Giguere was the MVP of the playoffs.
He is one of only four players remaining from that Anaheim team, which also has had a change of owners, front-office personnel and coaches. Rob Niedermayer, who had a goal and assist in the clincher against Detroit; Andy McDonald and Pahlsson are the only others still with the Ducks.
The team also dropped the "Mighty" from their name after Henry and Susan Samueli bought the franchise from The Walt Disney Co. in June 2005.
Two of Detroit's third-period goals against Giguere came on deflections, including Datsyuk's score with 9:52 left.
The Ducks scored once against Hasek on a deflection, and the other three on rebounds.
Getzlaf made it 3-0 on a power play at 18:33 of the second, scoring on a goalmouth scramble after the Red Wings were assessed a bench minor for having too many men on the ice.
Rob Niedermayer scored a short-handed goal on a deflection 3:51 into the game. Perry gave Anaheim a two-goal lead when he tapped in a rebound midway through the second period.
The second game of the Stanley Cup finals will be in Anaheim on Wednesday, May 30, then the series shifts to Ottawa for Games 3 and 4 on June 2 and June 4.
- The Ducks are 7-0 in series-clinching games at home, including all three rounds so far this year.
- All four times the Ducks and Red Wings have met in the playoffs, the winning team advanced to the Stanley Cup Finals. This was the first time they faced each other in the conference finals.
- The Ducks and Senators will be playing each other for the first time since Jan. 19, 2006 at Ottawa, when Anaheim won 4-3 in a shootout. Hasek was the Senators' goalie that night. The teams haven't played in Anaheim since Oct. 17, 2003, when the Ducks were beaten 3-0 by Patrick Lalime.