The European champions kicked off their Champions League defence by thrashing the Swiss side 5-1 at the Santiago Bernabeu.
Rodriguez put Real 4-0 up in the 37th minute after Gareth Bale and Cristiano Ronaldo had added to Marek Suchy's early own goal.
Derlis Gonzalez's goal was scant consolation for Basel, whose drubbing was completed by Karim Benzema after the interval.
Having lost two of their opening three Liga games this season, including a 2-1 home loss to Atletico Madrid, pressure has heaped on Real and coach Carlo Ancelotti, but a strong start in Europe delighted Rodriguez, who hopes the performance can be repeated starting with a trip to Deportivo La Coruna on Saturday.
He said: "I had a good game. Everything came off for us and as a team we did everything well.
"We hope to continue playing like this and keep scoring goals. This match will give us confidence for what's ahead."
"I also had a good game against Atletico but I think I was better today.
"I think Cristiano Ronaldo and I have always had a good relationship, we always talk to one and other and he's a great person.
"As for the group, we are mentally strong."
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