The 33-year-old from Czech Republic won 85% of points on his first serve, breaking his Ukrainian opponent - the world number 16 - four times as he cruised to a 6-3 6-1 victory in just 65 minutes to set up a second-round clash with Olympic bronze medallist Juan Martin del Potro.
Germany's Tommy Haas, beaten by Dolgopolov in the Citi Open final on Sunday, fared better than his Washington conqueror in his first-round match, responding from the disappointment of passing up four match-points in the second-set tie-break to defeat Argentina's David Nalbandian 6-2 6-7 (11/13) 6-3.
Americans Mardy Fish and Sam Querrey advanced with straight-sets wins, 11th seed Fish crushing Australia's Matthew Ebden 6-2 6-0 and Querrey dropped only nine points on serve en route to a 6-3 6-2 triumph over Austria's Jurgen Melzer.
Canada's Milos Raonic, the 16th seed, was in second-round action and progressed thanks to his first win on his third main-draw appearance at his home tournament - a 6-3 6-4 success against Viktor Troicki of Serbia.
Raonic's compatriot Vasek Pospisil missed out on a place in the last 16, though, going down 7-5 6-4 against seventh-seeded Argentinian Juan Monaco.
Also in the second round, American eighth seed John Isner overcame Spain's Pablo Andujar 7-6 (7/5) 7-5 in the late match.
In other first-round encounters on Tuesday there were wins for Russia's Mikhail Youzhny, Spain's Marcel Granollers, Italian Fabio Fognini, Frenchman Julien Benneteau and Kazakhstan's Mikhail Kukushkin.