De Bakker upset third-seeded Jo-Wilfried Tsonga of France, 6-4, 3-6, 6-3 for his first career win over a top-10 ranked opponent. Earlier, Fernando Verdasco and David Ferrer both won to set up an all-Spanish matchup in the other semi-final.
Soderling broke Schwank in his first service game and never looked threatened from there, using his powerful serve to wrap up the win in 68 minutes.
"So far so good this week," Soderling said. "It was tricky today with different conditions. It was quite cold and the balls are heavy, but overall I think I played a good match. I served well, managing to win quite a few easy points on my first serve."
De Bakker converted his only break point of the first set against the 10th-ranked Tsonga, but was then broken twice in the second as he struggled with his serve. He responded with two breaks of his own in the decider to become the first Dutchman since Richard Krajicek in 1994 to reach the semi-finals of the tournament.
"It feels great to get my first win over a top-10 player," de Bakker said. "It's a big step forward for me. I had a tough match with Jo in Davis Cup last year, so it feels great to come through this time."
The fifth-seeded Verdasco saved seven break chances late in the second set as he held on for a 6-2, 7-6 (4) win over Ernests Gulbis of Latvia, while Ferrer moved a step closer to reaching his third straight final after beating Thomaz Bellucci 6-4, 6-0.
"He's a top tennis player, especially on clay," Verdasco said. "It'll be a difficult but nice match. At least one of us will be in the final."
Verdasco, who lost Sunday's Monte Carlo Masters final to Rafael Nadal, broke Gulbis in the fourth and eighth games to take the first set before starting out 2-0 in the second.
But 44th-ranked Gulbis broke back before drawing level.
Both players held their serves until the 11th game, when Gulbis failed to convert seven break chances to move ahead. Gulbis threw his racket into the air in frustration after hitting his seventh chance long, and Verdasco finally took the set with a winner at the net.
Verdasco led 5-1 in the tiebreaker before clinching it on his second match point after Gulbis hit a backhand wide.
"I'm happy to be in the semi-final for the first time here," said Verdasco, who is 5-3 against Davis Cup teammate Ferrer.
The eighth-seeded Ferrer, who lost to five-time champion Nadal in his two final appearances, broke his Brazilian opponent four times to reach the semi-final for a fourth consecutive time.