Nadal, playing his first singles match since the U.S. Open fourth round and becoming a father, won the first set against American Tommy Paul but went down 3-6 7-6(4) 6-1.
Defeat ended 22-time Grand Slam champion Nadal's hopes of reclaiming the number one ranking from Alcaraz this week.
World number three Medvedev, who won the Vienna title last week, lost 6-4 2-6 7-5 to Australian Alex de Minaur.
Alcaraz was quick out of the blocks and struck 30 winners to overpower his Japanese opponent despite losing serve in the first set on the only break point he faced all match.
"First round in every tournament is never easy. You have to be really focused, you have to try to get good rhythm, good feeling in the first round," said the 19-year-old Spaniard.
"I had to get used to this court in the match. I felt (it was) fast, this court, but I'm a good player on indoor courts... these kind of courts help me with my game."
Medvedev fought back in the second set but De Minaur put on a gutsy display in the decider, coming from a break down to seal his first career win over a top-five player.
A frustrated Medvedev smashed his racket into the ground after double-faulting on match point before shaking hands with De Minaur at the net.
"It's the end of the year, everyone's a bit tired, but I'm very proud of my performance. I just played very smart," said De Minaur. "It was an absolute battle and I'm very happy I was able to come out on top."
The Australian next faces U.S. Open semi-finalist Frances Tiafoe who saw off unseeded Jack Draper 6-3 7-5.
Earlier, fourth seed Stefanos Tsitsipas produced a fine performance to oust Dan Evans, breaking the Briton early in both sets and making the most of his opponent's unforced errors to secure a 6-3 6-4 victory.
Canada's Felix Auger-Aliassime and Russian Andrey Rublev sealed the last two places at the season-ending ATP Finals in Turin as Taylor Fritz's hopes vanished with a loss to Gilles Simon.
Auger-Aliassime battled to a 6-7(6) 6-4 7-6(6) win over Swede Mikael Ymer while Rublev beat John Isner 6-2 6-3 on Tuesday.
They join Alcaraz, Novak Djokovic, Nadal, Casper Ruud, Tsitsipas and Medvedev in the event.
