The Dutch have never reached the competition's final since first participating 104 years ago but knocked out Rafael Nadal's Spain in the last eight on Tuesday.
Jan-Lennard Struff sought three-time winners Germany's hopes alive in the second singles match against Tallon Griekspoor.
Van de Zandschulp claimed his scrappy victory with his 10th match point in two hours 44 minutes.
"It could have been easier but I did it the hard way," said the Dutchman.
"At some point, I didn't know what to do any more on the match points."
Neither player blinked in a first set with few thrills, until the Dutchman, ranked 80th, nosed ahead for a 5-4 lead which he served out, clinching when Altmaier went wide.
Van de Zandschulp, the final player to beat retiring superstar Nadal when the Netherlands knocked out Spain, broke in the fifth game of the second set and consolidated for a 4-2 lead, taking full control of the match.
The Dutchman forced four break points in the next game, but the world number 88 managed to escape with three aces to avoid a double break.
Moving into a higher gear the German was able to get back on serve at 4-4, converting his third break point at the end of the match's longest rally when Van de Zandschulp failed at the net.
The Dutch player brought up five match points in the tie-break but could not hold his nerve and Altmaier took his fourth set point with a smash to force a third set.
Van de Zandschulp broke in the second game, but Altmaier immediately responded.
The Dutchman produced another break of serve to open up a 5-3 lead and, after wasting four more match points, wrapped up the win with his 10th as Altmaier could not return a powerful serve.
"I had the toughest match of my life on Tuesday (against Nadal), so everything that comes next is a little bit easier," added Van de Zandschulp.
