Hamburg squandered a two-goal first half lead and were reduced to 10 men just past the hour after a red card for David Rozehnal.
The visitors still thought they had the game won when Marcus Berg got his second to make it 3-2 with 10 minutes left.
However, Kuranyi, who had already scored with a header in the 50th minute, dashed their hopes with his second goal to leave Schalke in fourth place on 20 points.
Bayer Leverkusen, who drew 1-1 with Borussia Dortmund on Friday, have 22 points and lead Hamburg on goal difference. Werder Bremen moved into third spot on 21 after a 4-1 win at VfL Bochum, with Bayern Munich fifth on 18.
"We deserved to win because the team showed character," Hamburg coach Bruno Labbadia told reporters. "We could have decided the game then and there," he said of a missed chance by Jonathan Pitroipa for a 3-0 lead early in the second half.
Schalke had come close after 14 minutes when Halil Altintop launched a quick break, finding Peruvian Jefferson Farfan on the edge of the box whose shot was kicked wide by keeper Frank Rost.
Hamburg, still unbeaten this season, were more productive in the 26th when Eljero Elia left Peru defender Carlos Zambrano standing as he sprinted into the area, cutting the ball into the middle for Sweden's Berg who scored from six metres.
THUNDEROUS FREE-KICK
The visitors doubled their lead with the final kick of the first half, Piotr Trochowski thundering a free-kick from 25 metres into helpless goalkeeper Manuel Neuer's top corner.
But Schalke punished the visitors for failing to make it 3-0 immediately after the restart, following a fine individual run by Pitroipa, with two goals in 11 minutes.
Kuranyi rose above defender Guy Demel to head in a Rafinha cross before Hamburg defender Rozehnal was sent off for bringing down the striker outside the box.
Lukas Schmitz headed in the equaliser after Marcelo Bordon's powerful free-kick was punched on to the bar by Rost, amid wild celebrations from most of the 61,000 fans in the stadium.
Berg silenced the home crowd briefly when he put Hamburg back in front before Kuranyi gained a point for Schalke with his fifth league goal of the season.
"That is how I imagine passion, emotion and atmosphere," Schalke coach Felix Magath told reporters. "We clinched the draw with a lot of effort but you could tell we were still weak in some aspects of our game."
Bottom side Hertha Berlin ended an eight-match losing streak, grabbing their first point since August by drawing 0-0 with champions VfL Wolfsburg who are seventh on 17 points.
Hertha, who are four points adrift of the four teams about them in the standings, put in a spirited performance and forced visiting goalkeeper Diego Benaglio to make several fine saves.
