Riley Sheahan scored in regulation for the Red Wings, who won for the second time in their past six games (2-3-1).
"We've got to start winning some games with the standings so tight. This is a good start," Sheahan said. "We're not going to pick and choose how we win at this point of the season so long as we're coming out on top."
The Oilers (23-36-9) have lost three of four. Edmonton's penalty-killers did everything they could to win the game, killing off delay-of-game penalties late in the third period and again in overtime.
Ryan Smyth tied the game at 1-1 at 8:43 of the third period when he fired a wrist shot over Howard's glove following a defensive-zone turnover by Brendan Smith. The goal was Smyth's 10th this season and the 15th of his career against Detroit.
"I wish we would have got a better result. I thought our guys worked hard for it," Edmonton coach Dallas Eakins said. "Our penalty kill has been excellent."
The Red Wings were 0-for-4 on the power play. The Oilers were 0-for-2 with the man advantage.
Sheahan opened the scoring when he snapped home his fifth goal at 13:53 of the first period off a nifty passout from Tatar. The rookie center beat Oilers goalie Viktor Fasth over his glove from just below the right circle for his fifth goal.
Sheahan has 15 points in 26 games with Detroit. The goal was his first in six games; he had not scored since Feb. 27 against the Ottawa Senators.
"Riley Sheahan was a very good player for us tonight. Very, very good," Detroit coach Mike Babcock said. "He just seems to get better and better all the time."
Howard, who sat out the game Tuesday against the Columbus Blue Jackets with the flu, made 21 saves in regulation and overtime to win his first game in March and snap a personal three-game losing streak.
"He was awesome for us," Sheahan said of Howard. "I think he'll continue to be awesome for us down the stretch. He's an unbelievable goaltender."
Oilers center Boyd Gordon was called for delay of game for firing the puck over the glass at 17:06 of the third, but Fasth stopped rookie Teemu Pulkkinen on the Red Wings' only shot. Defenseman Martin Marincin was called for the same penalty at 1:21 of overtime, and Fasth made a great save on a slap shot by Daniel Alfredsson 1:10 into the penalty to get the game to the shootout.
"I didn't think we had good looks tonight, and we didn't enter the zone well," Babcock said of his team's performance on the power play.
The Oilers came to Detroit after a 6-2 road loss to the St. Louis Blues on Thursday, while the Red Wings rested. Still, Edmonton looked like the fresher team in the third period.
"I thought we had a decent first period and then in the second we lapsed a bit," Smyth said. "I thought we picked it up in the third. We played aggressive and kept them on their heels a bit."
Pulkkinen made his NHL debut for the Red Wings. He was called up from Grand Rapids of the American Hockey League to take injured Tomas Jurco's place in the lineup.
"He's gotta take better care of the puck, but he can sure shoot and he can really skate," Babcock said of the 22-year-old forward's debut.
The Red Wings (30-23-13) visit the Chicago Blackhawks on Sunday night. The Red Wings and Philadelphia Flyers each have 73 points, but the Flyers hold the second of two Eastern Conference wild-card spots for the Stanley Cup Playoffs entering Saturday by virtue of having played one fewer game. The Flyers begin a home-and-home series against the Pittsburgh Penguins on Saturday.
The Oilers wrap up a four-game road trip Sunday against the Carolina Hurricanes.
Material from team media was used in this report