Zverev limited his unforced errors, converted his two break point opportunities and saved four of the five break points he faced before closing out the one-hour and 44-minute contest with a forehand volley into the open court.
"It was a difficult match. I thought he was in control of it throughout the first set," said former Miami Open finalist Zverev. "I was just hanging on and sometimes that is just what you need to do."
Eubanks had a chance to serve out the first set at 5-3 but was denied by Zverev, who broke the American again early in the second set to build a 3-0 lead. Zverev later had to save three break points to serve out the match.
Awaiting Zverev in the last 16 will be 15th seed Khachanov, who converted his fifth match point to secure a 6-1, 5-7, 7-6(5) win over Argentine 20th seed Francisco Cerundolo.
Zverev has a 3-2 record against Khachanov and won their most recent meeting in the gold medal match at the Tokyo Olympics.
Spanish top seed Carlos Alcaraz, looking to take another step in his quest to complete the 'Sunshine Double' after winning the Indian Wells title last week, will kick off the evening session against Gael Monfils in a third-round clash.
