Vazquez's third goal in four games helped Cincinnati (7-8-7, 28 points) earn a point for the second time during team assist leader Luciano Acosta's suspension.
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Cincinnati remained seventh and in the final playoff spot in the Eastern Conference as it continued its pursuit of a first postseason berth.
Teal Bunbury opened the scoring with his first goal for Nashville and the 65th of his 13-year MLS career as Nashville (8-7-7, 31 points) earned at least a point for the eighth time in 13 away matches this season.
The Coyotes remained fifth in the Western Conference, and they'll play eight of their final 12 matches at home in search of their third consecutive postseason appearance.
Bunbury had the visitors in front after six minutes from a corner kick.
Nashville defender Walker Zimmerman redirected the initial corner toward goal, where goalkeeper Celentano saved it off his hip. But Bunbury was in the path of rebound, and he contorted his body to send an acrobatic bicycle kick over the goal line from close range.
Vazquez had the home side level seven minutes after halftime from another corner kick.
Alvaro Barreal tried to volley the initial corner toward goal out of the air from the edge of the penalty area.
It deflected off the legs of a Nashville defender but continued toward goal, where Vazquez snuck away from his defender and redirected a header past Willis.
Vazquez had the best chance at deciding the match only two minutes later when he made an excellent daring run in from the right flank. But the American curled his left-footed effort just wide of the far-left post.