Gareth Bale finished off a lovely move involving Luka Modric and Cristiano Ronaldo before Ronaldo and James Rodriguez tapped home from close range.
Derlis Gonzalez pulled a goal back for the visitors moments later, but Karim Benzema rounded off the scoring for Real with a thumping finish late on.
"The reaction of the team today was good," said Real boss Carlo Ancelotti.
"We pressed very well in the first-half and created a lot of chances. "Not everything is resolved just yet, but I liked the reaction of the team."
Basel coach Paulo Sousa meanwhile was left to rue his side's naivey in defence despite showing flashes of their quality on the ball.
"We created chances, we were brave to have the ball and didn't just look to defend.
"In certain moments we were too naive, but in other moments we showed good quality and personality."
Ancelotti resisted calls for widespread changes following his side's third defeat in six games this season with full-backs Marcelo and Nacho the only new faces from the side that started against Atletico.
However, Nacho had a big role to play in the opening goal on 14 minutes as he raced onto Rodriguez's lovely backheel and fired in a low cross that Suchy could only direct into his own net.
Basel skipper Marco Streller nearly levelled immediately as he drilled into the side-netting from a narrow angle.
However, Real were gaining momentum as the half wore on and Tomas Vaclik had to make fine saves from Modric and Pepe before the three-goal blitz arrived.
Bale registered his second goal of the season when he was set free by a wonderful through ball by Modric with the outside of his foot and the Welshman lobbed the ball over the advancing Vaclik before slotting home.
It was 3-0 moments later when Modric fed Bale once more and this time he squared for Ronaldo to finish.
And Rodriguez rounded off his most encouraging display since sealing a big-money move from Monaco as the World Cup's top scorer was left with a simple task after Vaclik could only parry Benzema's initial effort.
Basel stuck to their task impressively and responded with a goal of real quality as Luca Zuffi picked out Gonzalez's run and the Paraguayan arrowed a shot across Iker Casillas and into the far corner.
There were more encouraging signs for Basel at the start of the second half as Real's intensity dropped and Casillas had to make an important save to prevent Gonzalez scoring his second of the evening after an error from substitute Raphael Varane.
Bale should then have teed up Ronaldo for his double instead of going alone when clean through as Vaclik made another impressive stop with his feet.
Basel continued to play some impressive football and Fabian Schaer was desperately unlucky to see his piledriver come back off the post.
However, Real showed their quality in the final third 11 minutes from time when Benzema and Ronaldo exchanged passes before the Frenchman smashed a shot in off the crossbar.