Wust finished 0.08 seconds behind Achtereekte at the Gangneung Oval, with Antoinette de Jong winning the bronze medal.
The gold was Achtereekte's first-ever Olympic medal, while Wust's silver draws her level with Germany's Claudia Pechstein on nine career Olympic medals, the most by any speed skater.
"Incredible. It's an amazing feeling," Achtereekte told a news conference. "I hoped. I was dreaming about it but I didn't expect it."
The 28-year-old, who will not race again at the Games, was in the fifth of the 12 pairs to take to the tracks and faced an agonising wait to see whether her time would be enough for victory.
"It was hard to wait," she said. "It was really close. (Wust) stayed under my time for really long. It was only the last lap that I won it and that's my power.
"I stay really flat and that's also my plan. Ireen started really fast but I did also. I did a really flat race and that's what won it in the end."
Wust, who will retire after the Pyeongchang Games, stormed out of the gates and was ahead of Achtereekte and seemingly headed for victory until the final corner, when she slowed fractionally to just miss out on gold.
"I just went for it," a disappointed Wust said. "I think I had a really strong beginning of the race. I raced for the gold, not for silver or bronze.
"I went really well, only the last lap was really hard. In the end I was so close but it's not good enough. I have to be satisfied with the silver, although it hurts a little bit."
If Wust had won, she would have become the first Dutch athlete to get five Olympic gold medals, but the 31-year-old will have another shot at that target when she lines up for the 1,500m on Monday.
"My dream was to have four Olympics -- Turin, Vancouver and Sochi I won gold, and to win gold again here in Pyeongchang but I have another chance on Monday. I'm looking forward to that," she said.
"I'm always determined for gold but maybe now a little extra."
(Editing by Clare Fallon)
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