Draper fired nine aces to Alcaraz's one, but Alcaraz saved four of seven break points and converted four of nine break-point chances in order to advance.
Fourth seed Marin Cilic of Croatia was less fortunate, losing to French qualifier Arthur Rinderknech 6-4, 3-6, 7-6 (5).
Alexander Bublik of Kazakhstan and Brandon Nakashima of the US also won their opening matches.
Erste Bank Open
Fourth seed Taylor Fritz smashed 21 aces to fight his way past Japanese qualifier Yoshihito Nishioka 6-7 (4), 7-6 (7), 6-3 in the first round in Vienna, Austria.
The American had 59 total winners to overcome 39 unforced errors. He won 57 of his 70 first-service points (81%), saved four of five break points and rallied from behind 5-4 in the second-set tiebreaker.
No. 3 seed Andrey Rublev of Russia coasted past Argentina's Diego Schwartzman 6-4, 6-1. Rublev had seven aces, saved all three break points he faced and broke Schwartzman's serve four times.
Croatia's Borna Coric, Russia's Karen Khachanov and Argentina's Francisco Cerundolo also won Monday.
