Alonso jousted his way up from 11th on the starting grid to claim an emotional victory for the Spaniard in front of his home fans after pole sitter Sebastian Vettel's Red Bull stalled on the track midway through the race.
Kimi Raikkonen of Lotus finished second, while Schumacher completed the podium after Lewis Hamilton and Pastor Maldonado collided on the penultimate lap as they fought for third place.
"This race is probably the best victory I have ever felt," said Alonso, who cried on the podium as the Spanish national anthem was played. "In terms of emotions, nothing compares to this."
Vettel's Red Bull teammate Mark Webber also had a strong showing, working his way up all the way from 19th to finish fourth.
Alonso's 29th career victory lifted him into first place in the points standings, 10 points ahead of Webber, who passed Hamilton and Vettel.
Force India's Nico Hulkenberg came in fifth and Schumacher's Mercedes teammate Nico Rosberg finished sixth. Paul Di Resta of Force India, McLaren's Jenson Button, Sergio Perez of Sauber and Maldonado in his Williams completed the top 10.
Alonso clinched an unlikely victory by passing a number of drivers on Valencia's 5.4-kilometer (3.3-mile) street circuit notoriously tough for overtaking. In 2009, the tight circuit that weaves through the America's Cup harbor didn't allow one single change in the order of cars from start to finish.
To sweeten the victory, Alonso saw both Hamilton and Vettel â€" his main rivals for the championshipâ€" fail to collect a single point.
After a poor qualifying session that left him out of the top 10, Alonso had said he had no chance at victory and had set a goal of simply getting as many points as possible.
But a combination of audacious driving and an eventful 57-lap race full of car trouble and crashes produced the most scintillating round of the already enthralling F1 season that had seen seven different winners in the first seven races.
"Sport is about that, up and downs. Yesterday, we had a sad moment but you can never give up," said Alonso. "In sport in general, but especially in Formula One, anything can happen. There is the car, the strategy, the overtaking, the maneuvering."
Alonso had an excellent start on a set of fast soft tires, jumping three spots to eighth place by the second turn. He then made short order of Webber, Bruno Senna and Schumacher to go fourth behind Hamilton.
"We had an amazing race, an amazing start. I remember six or seven overtakings we had," said Alonso. "Today we had all the good factors and luck with us."
Even so, victory for Alonso looked more than unlikely as Vettel, who had won here from pole the past two years, opened up a 20-second lead.
But following the appearance of the safety car, his Red Bull, which had been running impeccably all weekend, suddenly stalled on lap 34 and left him stranded hopelessly while the pack rushed by with Alonso at the head.
"Obviously it is a shame. We are not sure what happened. It is frustrating, we were en route to win the race," Vettel said. "We had a mechanical problem. The failure is still not clear, the car switched off."
Romain Grosjean had maneuvered to within two seconds of Alonso by lap 41 when his Lotus also sputtered to a stop, spoiling his good start.
With the French pilot out of the running, Hamilton found himself in second.
But the former champion couldn't cut into Alonso's advantage and as his pace faltered, Maldonado tried to pass and the two drivers collided, sending Hamilton's McLaren into the wall.
Schumacher slipped ahead of the pair to grab his first podium appearance since he returned to F1 in 2010 following a three-year hiatus for the seven-time champion.
"I didn't think about a podium, not even at the end of the race," said Schumacher. "At the end I asked the guys where we finished and the boys told me that's third. I can't believe that, I didn't expect it."
