Recent call-up Joey MacDonald started the second period in the Detroit net.
Detroit made it a one-goal game 2 minutes into the third as Henrik Zetterberg fed it to Bertuzzi from behind the net and he whacked at the puck under Dubnyk.
The Red Wings tied the game 3-all only 3 minutes later as Helm got the puck through to Miller in front of the net and had an easy redirect for his 10th of the year.
Detroit took a 4-3 lead with just under 4 minutes to go as Bertuzzi showed some skill, passing it to himself through defenseman Andy Sutton's legs before beating Dubnyk up high.
With Dubnyk on the bench, the Oilers sent it to extra time as Eberle found the puck in a scramble in front and put it past MacDonald.
"It's disappointing to lose the extra point when we had the lead with under a minute to play," Red Wings captain Nicklas Lidstrom said. "We came back from a 3-1 deficit to tie the game up and then took the lead. So it's tough to take when you lose after battling back like that."
Notes: It was the third of four meetings between the two teams this season. Detroit won the first two games by 3-0 and 3-2 scores. The Oilers came into the game with seven straight losses to the Red Wings. ... Detroit G Jimmy Howard was out with a broken finger. ... Dubnyk made his career-high fifth consecutive start in goal for Edmonton. ... After an injury-filled season, the Oilers are almost back to full strength as the league's rookie points leader, Nugent-Hopkins, returned to the lineup after missing the last 13 games with a shoulder injury. D Tom Gilbert, the only Oiler still on the IR, is expected to play early next week. ... Bertuzzi was able to play after being listed as questionable going in with a charley horse.
PHT: A comeback win in Game 3 gave Ottawa hope, and the Senators seek a repeat result tonight to level the series.
PHT: The Canucks parted company with head coach Alain Vigneault and two assistants, according to a report.
Video: NHL from NBC Sports
Sutton officially retires
Steve Mears runs down the latest news in the NHL. Edmonton Oilers defenseman Andy Sutton has announced his retirement after 15 seasons in the NHL.
Check out photos from the Kings' Stanley Cup victory parade in LA.
Stanley Cup playoffs
Check out photos from the NHL postseason action.
Stanley Cup winners
A look at the teams that have earned the right to hoist Lord Stanley's prize since 1965.
Some of Hollywood's hottest celebrities take in NHL games.
Check out the ice girls from around the National Hockey League.