That let Djokovic, above all, and everyone else, know that Brooksby a 20-year-old wild-card entry from California who is ranked 99th and never before had been on this sort of stage belonged.
And then, not surprisingly, Djokovic showed why he is who he is and how he's managed to move within three victories of the first calendar-year Grand Slam in men's tennis in 52 years.
Settling in and sending messages, to the fans with roars and to Brooksby with some staredowns, the No. 1-ranked Djokovic improved to 25-0 in majors this year by winning 1-6, 6-3, 6-2, 6-2 on Monday night.
He extended his pursuit of a true Grand Slam and a record-breaking 21st major championship, while also making this the first time in the history of the U.S. Open, an event that dates to the 1880s, with zero men or women from the host country in the quarterfinals.
Was a good finish. It wasn't a good tart, said Djokovic, who hopes to add a fourth U.S. Open title to a collection that's been boosted by his ninth at the Australian Open in February, second at the French Open in June and sixth at Wimbledon in July.
He next meets No. 6 Matteo Berrettini of Italy in a rematch of the final at the All England Club.
One key to Monday's turnaround: Djokovic broke in Brooksby's initial service game in each of the last three sets.
I wanted to wear him down, Djokovic said, and it worked.
Affected by a left hip that bothered him earlier in the tournament, Brooksby was visited by a trainer after the second set and again after the third. Still, for someone who never had set foot on Ashe's blue court until about two hours before the match, when he got a chance to practice there, Brooksby never seemed overwhelmed by the setting or the circumstances.
We're going to see a lot of him in the future, said Djokovic, who would be the first man since Rod Laver in 1969 to win all four Slam trophies in one year.
I mean, he has the means. Now, obviously a lot of things have to come together. But he has a bit of an unorthodox game. Does a lot with the feel. He's smart. He knows how to win points.
Brooksby's 6-foot-4 strides and reach, his anticipation, his variety that includes a well-disguised two-handed backhand slice, his think-steps-ahead point construction, his commitment to patterns drawn up by his coach at home in Sacramento since age 7, Joe Gilbert all left Djokovic a tad discombobulated early.