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Australian Alex de Minaur battled back to triumph 4-6 7-5 6-1 against Jaume Munar.
He will play either Lloyd Harris or Grigor Dimitrov, who are scheduled to do battle on Tuesday (AEST).
Denis Shapovalov - the highest seed in action on Day 1 - relished feeling much better on court as he impressively bounced back from two consecutive defeats to brush aside the challenge of Dusan Lajovic, winning 6-1 6-3.
Shapovalov had suffered disappointment when he was knocked out in the second round of the Estoril Open, an event where he was the number one seed.
He has not reached the quarter-finals in his previous two ATP Tour events but won 90 per cent of his first-serve points against Lajovic, hitting 21 winners to his opponent's five.
"Just definitely felt a lot better - I felt very good on the court," Shapovalov said after a 63-minute win in which he did not lose serve. "I'm just happy to get the first win.
"Not much can change in the span of a couple of days, but Madrid is a place where I have had a lot of success, from back in juniors to making semis here in the Masters [1000 event] and finals of Davis Cup.
"It's just a place where it kind of feels like home to me and I just feel so comfortable."
Shapovalov will face Alexander Bublik – who saw off Marton Fucsovics in straight sets – in round two.Tommy Paul, meanwhile, will play Andrey Rublev after defeating Pedro Martinez.