At number 99 the lowest ranked player to make it through to the last 16, Tsurenko represented a tricky test for last year's U.S. Open runner-up having knocked off 12th seed Dominika Cibulkova and 21st seed Irina-Camelia Begu to get to the fourth round.
The 27-year-old Ukrainian looked prepared to add another seeded scalp to her Flushing Meadows collection as she forced the first set to a tie-break before surrendering 7-5.
After a draining 67-minute opening set, Tsurenko had little left in the tank for the second and Vinci stamped her authority on the match quickly by going up a break and cruising to an easy win to clinch a spot in the last eight for the third time in four years.
Next up for Vinci is the winner between second-seeded Australian Open champion Angelique Kerber and 14th-seeded twice Wimbledon champion Petra Kvitova.
(Editing by Larry Fine)
