The ATP Masters 1000 event's organisers were chastised after only one of the nine wildcards went to a Spaniard, with Draper of the United Kingdom being one of those who got straight into the main draw.
Draper, who also advanced to the second round of the Miami Open a fortnight ago, followed up his choice with his first-ever ATP triumph on clay, defeating world No. 27 Sonego 6-4, 6-3 in one hour and 38 minutes.
Draper has won four ATP Challenger events this year, the most recent of which was in April at Saint-Brieuc, France, and he continued that success into Madrid.
Draper broke in game four, but the Italian pulled level in game nine, only for Draper to break again and close off the set in the next game.
In the second set, there was just one break of service, with the pivotal game coming in game four, when Draper secured his second-largest ranking victory. In 2021, the then-world No. 19 Jannik Sinner was defeated by the 20-year-old at Queen's Club.
The British teenager will face sixth seed Andrey Rublev in the second round, but his first-round victory has already propelled him to a career-high ranking of No. 121 in the ATP Rankings.