Spezza beat Vesa Toskala with a shot on Ottawa's first attempt. Alfredsson also shot and scored on the Senators' third attempt to give Ottawa its second straight win.
Alex Auld stopped Nikolai Kulemin and Lee Stempniak in the shootout after turning aside 25 shots during the game.
Mike Fisher returned to the Senators' lineup and opened the scoring early in the first. Kulemin drew Toronto even moments later.
Toskala stopped 34 shots as the Maple Leafs earned a standings point despite extending their losing streak to five (0-3-2).
Ottawa outshot Toronto 35-26 overall, including a 5-0 margin in overtime as Toskala came up with a pair of big stops on Fisher in the dying seconds of the extra period.
The Maple Leafs ended a team-record 17-game stretch without being outshot.
The Senators hadn't played since their 4-1 win over the New York Rangers on Saturday ended a 0-4-2 skid.
Jason Blake did not play for Toronto because of a suspected concussion. Blake was injured when he was hit by Atlanta's Colby Armstrong on Tuesday night.
Fisher, who missed two games because of a knee injury, got credit for his third goal of the season 3:54 in when his centering pass toward Shean Donovan went into the net off Toronto defenceman Tomas Kaberle's skate.
Kulemin tied it 59 seconds later when he tapped in Mikhail Grabovski's pass through the crease for his fourth goal.
Ottawa left wing Dany Heatley showed his frustration after Toskala stopped his slap shot 12 minutes into the middle period when he skated over to the left side and swung his stick, slamming it hard against the boards.
The Finnish goalie also got into the Senators' heads early in the third when he got a piece of Chris Kelly's shot with his stick before it struck the left post and stayed out.
Senators pest Jarkko Ruutu drew a goalie interference penalty 13:40 into the third when he fell over Toskala in the crease after the goalie made a sensational save on Jesse Winchester's short-range shot.
Notes: Donovan left the game in the second period with an upper body injury and did not return ... A sellout crowd of 19,703 watched the game at Scotiabank Place ... The Senators will attempt to run their win streak to three on Saturday, when they face the Islanders in Uniondale, N.Y. (7 p.m., no TV, Team 1200).