Alcaraz struck a down the line forehand winner before double faulting to lose serve.
Serving to stay in the set, Norrie struck a down the line forehand winner to lead 15-0.
Alcaraz netted two down the line forehands to trail 40-15.
The Spaniard netted a down the middle backhand return to level the set a five games all.
Serving at 30-15, Alcaraz struck an overhead winner which he followed up with a down the line forehand winner to hold serve and lead 6-5.
Norrie served up gamepoint before Alcaraz struck a crosscourt forehand dropshot winner to reach deuce.
Norrie then netted a crosscourt backhand to face matchpoint.
Alcaraz struck an inside out crosscourt forehand dropshot winner to end the match after one hour and 33 minutes of play.
Norrie had used his left-handed down the line forehand effectively during the match but it would not be enough.
Today in round two on the clay courts of the Rio Open Presented By Claro, in Rio de Janeiro, Brazil the top seeded Alcaraz faces the veteran Fabio Fognini of Italy not before 2:00 pm PST USA.