Alcaraz saved three set points before clinching a see-sawing opening set with a strong display in the tiebreak but continued to experiment with his new sped-up service motion and was broken twice early in the next set.
The 23-year-old Spaniard, who had shown glimpses of his best form en route to the US Open quarter-finals last month after his long injury layoff, recovered to move ahead but a double fault in the seventh game handed Michelsen a route back.
Michelsen capitalised and kept his composure in a tense finish to the second set where a fortunate netcord winner left Alcaraz stranded at the baseline and helped the 22-year-old American force a decider.
A couple of breathtaking points gave Alcaraz a 2-0 lead in the final set and the seven-time Grand Slam champion used the launch pad to close out a comfortable victory on a day of big upsets in the ATP 500 tournament.
The top seed's win came after third seed Taylor Fritz fell to qualifier Jaume Munar 6-7 (7), 6-4, 6-3, sixth seed Casper Ruud lost to Luciano Darderi 7-6 (3), 6-3 and seventh seed Tommy Paul suffered a 6-1, 6-2 defeat by Alejandro Tabilo.
Holger Rune, the former world number four who is stepping up his comeback from an Achilles injury, also crashed out following a 6-3, 6-1 loss to Kyrian Jacquet.