Murray booked his spot on Tuesday night.
Norrie and Evans both joined Andy Murray in the third round to make it the first ever ATP Masters 1000 event on clay where Britain has had three players at that stage.
Murray booked his spot – and a meeting with old rival Novak Djokovic – on Tuesday night and Norrie and Evans followed suit the next day.
British number one Norrie beat big-serving American John Isner in three sets, winning 6-4 6-7 (5) 6-4.
The 26-year-old, ranked 11th in the world, was able to find decisive breaks at key moments of the first and third sets to make it through and set up a date with rising star Carlos Alcaraz.
Evans, who has been short of wins lately, posted an impressive 6-3 5-7 7-6 (2) success over home favourite Roberto Bautista Agut and will next play world number eight Andrey Rublev.