It was the first tournament the 19-year-old played as world number one, something he became after winning the US Open.
Alcaraz then became the youngest ever to become world number one since 1973.
- I couldn't handle the pressure he put on me and it's obviously something that I have to learn from and it became a lesson, says Alcaraz according to ATP's website.
31-year-old Goffin took his biggest win of the season.
In the qualifiers, he burned two match balls against Luca Nardi on Sunday, but still got a place in the tournament as a lucky loser and was able to win in the first round even though the world number one was on the other side of the net.
"Have faith in myself"
- I always have a belief that I can create problems for this type of player even though I didn't play my best tennis at the end.
I have kept faith in myself and been prepared to take the chance, says 66-ranked Goffin.
After the US Open, Alcaraz has played two matches in the Davis Cup organized by the International Tennis Federation.
The Astana Open tournament is extremely high class.
Seven of the world's ten best players have started.
Meanwhile in Tokyo, world number two Casper Ruud (who lost the US Open final to Alcaraz) lost in the first round to Spaniard Jume Munar 6-3,6-3.