Gilbert Brule and Sam Gagner also scored for the Oilers, who went 0-11-2 in their previous 13 games. Edmonton was one loss shy of tying the franchise-worst winless streak of 14 games (0-10-4), set in 1993.
"It's obviously a great feeling," Gagner said. "You never want to be in that situation. But it was great to have fun again. It's only one win. Hopefully it gives us some confidence."
The Oilers have just two victories in 22 games and are five points behind Carolina and Toronto for last place in the NHL. Edmonton hadn't won since beating Toronto at home on Dec. 30.
"It hasn't been easy, and it was certainly not something I expected to happen to this hockey team," Oilers coach Pat Quinn said. "We've had some rough spots in a lot of these games and haven't held up.
"We had a couple of rough stretches again tonight but we bent and came back. We had a pretty good outing. It's something we have to keep building on."
"It wasn't a pretty game to watch from either side," Carolina coach Paul Maurice said. "The puck was bouncing around a bit out there. I thought we were a bit sluggish out of the gate. We got better as the game went on.
"Cam [Ward] has really been the difference for us the last two weeks but he wasn't at his 'A' game best. But we have to put more than two in, as well."
Carolina scored on its first shot after winning a draw deep in the Oilers zone during 4-on-4 play. Alberts scored from the point, beating goalie Jeff Deslauriers.
The Oilers (17-31-6) tied it 5 minutes later when Jacques took a bad-angle shot that goalie Ward directed into his net with his glove.
Edmonton took a 2-1 lead with five minutes remaining in the first period as Andrew Cogliano took the puck out of the Oilers zone and sent it to Patrick O'Sullivan, who quickly directed it to Brule for his 12th.
The Hurricanes tied it again midway through the second period on the power play as Jokinen skated through the Oilers defense and flipped a soft shot over Deslauriers.
The Oilers went ahead 3-2 as Pouliot came out from behind the net and banked the puck off of Ward's pads and in for his first goal of the season.
Edmonton made it a two-goal game with a power-play goal with 3 minutes left in the second. Ward allowed a big rebound on a kick save on Mike Comrie's shot, and Gagner was there to send the puck into the empty net.
The Hurricanes created good pressure in the third, but the Oilers held on for their first win in 2010.
Quinn was pleasantly surprised by the rousing cheer from Oilers fans when the game ended.
"After all this, Edmonton fans are really something," he said. "The reception that the players got after a month of frustration was just awesome. That is one of the good feelings."
The teams hadn't met since Nov. 1, 2008, when Edmonton won 3-1 at Carolina. ... With D Sheldon Souray out with a broken hand after a fight with Calgary's Jarome Iginla on Saturday and fellow defensemen Steve Staios and Ladislav Smid sidelined by concussions, rookie Alex Plante made his NHL debut. He earned his first point with an assist in the first period. ... Carolina was without forwards Scott Walker, Tuomo Ruutu and Erik Cole, who missed the game because of upper body injuries. ... Ward, who hails from nearby Sherwood Park, stopped 129 of 134 shots over his four previous games.