Joffrey Lupul collected the disc in the defensive zone and raced up the left wing on a 3-on-1. Lupul held the puck until he got to the circle and feathered a pass over to Connolly, who easily tapped it in for the winner.
Ben Eager, Taylor Hall and Jordan Eberle lit the lamp for the Oilers, who had their four-game winning streak on home ice snapped. Nikolai Khabibulin was tagged for all four goals on 26 shots in defeat.
The Leafs wasted no time opening the scoring, finding the back of the net just 40 seconds into the contest.
Phil Kessel gained the blue line on the rush and cut to the slot before dropping it off for Lupul, who ripped a quick wrister past the glove side of Khabibulin.
Toronto extended its lead to 2-0 just 58 seconds later when Cody Franson sent a pass right up the middle to Lombardi at the blue line and the veteran center went in on Khabibulin and buried a backhander.
The Oilers cut the deficit to 2-1 54 seconds later after Lennart Petrell chased down a loose puck behind the net and swung it out front to Eager, who fired it into an empty cage.
Edmonton knotted the game at two at 8:15 of the second period when Hall drove hard to the net and was rewarded with a nice pass from Sam Gagner before redirecting it past Reimer from in close.
The Leafs retook the lead with a power-play tally at 6:46 of the third period after Clarke MacArthur backhanded a pass off the left wing wall right to Gardiner, who one-timed it past Khabibulin from the point.
Edmonton, though, fought back to tie things at three with 4:04 to play in regulation after Eberle fought off two checkers in the slot and fired a perfect wrister high over the left shoulder of Reimer and just under the crossbar.
"We had a slow start but came back hard and played a good game," Eager said. "We had our chances to win, but unfortunately couldn't convert."
Toronto completed a season sweep of the Oilers after taking the first game of the series, 6-3 on February 6...It was Connolly's first goal since January 5...The Leafs improved to 4-3 in overtime...Toronto is 19-3-0 when leading after the first period...The Oilers finished 0-for-2 with the man advantage, while the Leafs were 1-for-3 on the power play...Toronto will face the Vancouver Canucks on Saturday, while Edmonton will host the Colorado Avalanche on Friday.