Achtereekte edged out defending champion Ireen Wust by 0.08 seconds to claim the gold, with Antoinette de Jong taking the bronze. The Dutch will be hoping for more speed skating clean sweeps at PyeongChang 2018, having recorded four in Russia four years ago.
Racing in one of the early pairs, Achtereekte set a challenging target of 3:59.21. The 31-year-old Wust, who will retire after the Games, looked set to beat it only to lose momentum coming out of the final corner and just miss out on a fifth Olympic title.
"A gold medal, it's an incredible feeling," said the 28-year-old Achtereekte on claiming her first Olympic medal. "I cannot believe it. It was such a good race. I skated so well. I've always skated the 5,000m and I've been getting better at the 3,000m and I managed to qualify for this one. I feel so good and I thought I'd have to skate the best race of my life, which I did. It's incredible."
Expressing her feelings at just missing out on another Olympic title, Wust said: "I'm feeling OK, a little disappointed. I was on schedule for the gold but the last lap was too hard and I didn't make it.
"I have to be happy and I will be happy with silver, but I came for gold so it's a little disappointing. It was a hard one. I had a really strong race today, but Carlijn was the best today. She is the champion now and I have to be happy with silver."