Image credit: PA Sport
Briton Dart, 22, lost two consecutive service games in the first set to the world number 11 and won only 12 points in the final set.
Eighth seed Sevastova, who will go on to play Naomi Osaka in the next round, converted on her fourth match point.
At the same tournament, world number seven and fourth seed Petra Kvitova suffered a straight-sets defeat to Estonia’s Anett Kontaveit 7-5 7-6 (1).
A break in the 11th game of the first set was enough for the world number 20 to draw first blood, while four breaks of serve in the second set saw it finish 6-6.
Kvitova was broken four times, the last as her opponent had match point.
At the ASB Classic in Auckland, top seeded 2017 WTA Tour finals winner Caroline Wozniacki beat Laura Siegemund of Germany 6-3 6-2 and number three seed Su-Wei Hsieh saw off Monica Puig 6-1 7-6 (3).