Hassan picked herself up after getting in a tangle with Kenyan runner Edinah Jebitok at the start of the last lap. She sped around the outside of the pack on the back straight and ended up crossing the line first in 4 minutes, 5.17 seconds to qualify for the semifinals.
It kept alive the Dutch runner's bid for a rare distance-running treble at the Tokyo Games.
Hassan qualified to run in the 5,000-meter final later Monday at the Olympic Stadium, where the energy she expended on that last-lap scramble in the 1,500 heats might catch up with her.
She's expected to battle with two-time world champion Hellen Obiri of Kenya for the 5,000 gold.
Hassan, who won a 1,500- and 10,000-meter double at the world championships in 2019, says she will also run in the 10,000 meters at the Tokyo Games and go for three medals.
That grueling schedule means she will have to run six races in eight days in the Tokyo heat and humidity, including the two races on Monday and the 1,500 and 10,000 finals on back-to-back days on Friday and Saturday.