Milan Lucic and Klefbom snapped 29- and 33-game goal droughts, respectively. Jujhar Khaira and Jesse Puljujarvi scored their 11th goals of the season each and Ryan Strome recorded his 100th career assist.
"That's a heavy weight that both of them have been carrying for two months," said Oilers Head Coach Todd McLellan about the busted slumps for Klefbom and Lucic.
Connor McDavid assisted on the game-winning goal, putting him up to 50 assists and 80 points this season.
Cam Talbot stopped 32 of 35 shots, finishing the night with a .914 save percentage.
The Oilers went 3-for-3 on the penalty kill and 1-for-4 on the man-advantage.
Edmonton saw yet another two-goal lead evaporate late in the game but rallied in the overtime period to win 4-3.
"It was a night of ups and downs and some inconsistencies but we found ways to recover," said McLellan.
It was a goal-filled first period, with the Oilers striking in all situations: shorthanded, even strength and on the power play.
First, Coyotes defenceman Jason Demers opened the scoring at 3:29. The rearguard fired a shot from the blueline that redirected in off Kris Russell's skate.
"We've battled that first goal against all year," said McLellan. "That shot is going at least 12 to 15 feet wide and off a skate and in. That was a bit of a downer but then how were we going to react? Were we going to roll over and feel sorry for ourselves? We didn't."
Then, it was all Oilers.
Khaira evened it up shorthanded while killing Anton Slepyshev's hooking minor. Strome and Khaira found themselves on a 2-on-1 during the penalty. Strome elected to keep and shoot, with the rebound popping out to Khaira who put it past Darcy Kuemper.
Lucic breathed a heavy sigh of relief at 13:21 by scoring his first goal since Dec. 23 with a howitzer at even strength. The winger walked in and released a slap shot far side on Kuemper to net his 10th of the season and 32nd point.
"Nice to have one finally go in," said Lucic. "It creates some positivity with your mindset and you hope you can build off that."
Edmonton scored their third marker while on the power play. Puljujarvi received a nice feed from Ryan Nugent-Hopkins to snipe his 11th of the year, giving the Oilers a 3-1 lead heading into the middle stanza.
Arizona pushed in the second period, notching the first six shots on goal while keeping Edmonton from generating much offensively. Talbot stood them all up, however, stopping Max Domi, Brendan Perlini and an awkward hop off a defensive zone draw.
Talbot ended the frame with 12 saves to take the two-goal lead into the final 20 minutes of play, where it was erased.
The Oilers got close to adding an insurance marker on the power play, as Draisaitl dinged a one-timer from Ethan Bear off the iron. Then came the late push from the 'Yotes.
Coyotes forward Christian Dvorak made it 3-2 at 11:31. Dvorak received a pass from Richard Panik and slammed the puck short side on Talbot.
Shortly after the goal, Michael Cammalleri was called for tripping, causing the Oilers to kill a penalty with less than 10 minutes to go. The 'Yotes nearly tied it on the man-advantage, but a deflected shot that Talbot couldn't find went wide of the net.
The Oilers killed the penalty but Niklas Hjalmarsson tied it with under five to play. A scramble ensued in front of Talbot and the Oilers were unable to clear the puck. Hjalmarsson intercepted a clearing attempt and whipped a shot in for his first of the season.
"It's not good enough," said Klefbom of his club's inability to maintain two-goal leads. "We've got to be better, got to be smarter and make better decisions when we're up two goals."
Draisaitl hit the post with under a minute left but the two sides proceeded to overtime where Klefbom tallied his first goal in 33 games to win it.
"It was very nice for me to finally score a goal," said Klefbom. "I'm very fortunate to be able to play with them (Draisaitl and McDavid) at 3-on-3. It's a lot of fun."
The Oilers continue their five-game home stand with a date against the New York Islanders on Thursday.