The 30-year-old Australian, who qualified for the main draw with two straight-sets wins - including a 6-0 6-1 thrashing of compatriot Alex Bolt - defeated Zachary Zvadja 6-3 6-1 on Monday (Tuesday AEST).
Vukic, the world No.105, lost in the first round on grass at Eastbourne, Wimbledon and Newport, before winning five matches to claim the event on the second-tier Challenger circuit in Granby, Canada.
But the American world No.73 had won the previous three matches in straight sets between the pair.
The lucky loser was no match for Vukic this time though, rolled in just 51 minutes having his serve broken twice in each set.
Next up for the Sydneysider is world No.15 and fourth-seeded Italian Lorenzo Musetti.
Fellow Australians Alex de Minaur, the top seed and defending champion, and teenager Cruz Hewitt will also play singles, along with Daria Kasatkina in the women's draw.
World No.6 de Minaur returns to the American capital 12 months after beating Spain's Alejandro Davidovich Fokina to claim his 10th ATP title and for his first event since marrying British No.2 Katie Boulter.
The Demon begins his defence against two-time grand slam finalist Stefanos Tsitsipas, and victories for de Minaur and tour-level debutant Hewitt will see the pair meet in round two.
It is close to 30 years since Lleyton Hewitt first appeared at the ATP 500 tournament, and his son has qualified for the main draw after Svajda and Dusan Lajovic both retired when trailing the 17-year-old.
The Wimbledon boys' singles finalist, ranked 607th, will next take on American Marcos Giron, aiming to record his third straight top-200 win.
Elsewhere, there were opening wins for Californian Taylor Fritz, the world No.10 and third seed, and 30th-ranked Chilean Alejandro Tabilo.
Two-time singles champion Nick Kyrgios will return to the event later in the week, featuring in doubles, as will compatriots Marc Polmans and John-Patrick Smith.
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