Medvedev has now won 22 of his past 23 matches, a remarkable run that includes titles in Rotterdam, Doha and Dubai and a runner-up finish at Indian Wells.
World number 119 Eubanks, who only dropped one set en route to the quarter-finals, looked comfortable early on as he held to love to start the match and led 3-2 before play was interrupted by a 30-minute rain delay.
When the players returned to the court, Medvedev asserted his dominance with a quick hold to love, took advantage of the first break point opportunity he earned to grab a 4-3 lead and closed out the set with another break.
In the second set, Medvedev looked to have settled into a groove as he dropped only three points on serve to move ahead 3-2 and roared back from 40-15 down to break for a 4-2 lead.
But Eubanks quickly got the set back on serve when he became the first player of the Miami fortnight to break Medvedev and levelled at 4-4 before the Russian held to love to put the qualifier on the brink.
Eubanks again drew level but Medvedev delivered another love hold and, with the finish line in sight, broke the American for the fourth time to seal victory on his third match point when his opponent sent a forehand smash wide.
Medvedev next faces fellow Russian Karen Khachanov (26), the 14th seed, after he beat Argentine Francisco Cerundolo (24) 6-3, 6-2.
