Elsewhere, a weakened Barcelona were comfortable 2-0 winners at third-tier Ceuta with Maxwell and Pedro netting first-half goals.
La Liga leaders Madrid were keen to avoid a repetition of last year's shock exit to Segunda B Alcorcon, who dumped Manuel Pellegrini's team out at the same stage 4-1 on aggregate.
The attitude of former coach Pellegrini's side was questioned after their 4-0 first-leg defeat and in the build-up to Tuesday's game Mourinho warned a repeat performance would see players dropped for good.
"There was no 'Alcorcon shock'. We played a serious game and the result, bearing in mind it is a knockout competition, is good," the Portuguese told reporters.
"I would sign for the same in every Champions League away leg. With the attitude I saw tonight it won't be a problem to reach the next stage. I'm sure we'll win the second leg."
Mourinho fielded a changed side at the Nueva Condomina but still included the likes of Iker Casillas and Cristiano Ronaldo and they dominated against Murcia, who like Madrid are unbeaten this season and top their league.
France striker Karim Benzema was given a rare start in place of Gonzalo Higuain.
Mourinho had challenged Benzema to prove his worth but he was substituted on the hour after an ineffective display.
"It's clear some players haven't had a great game but everyone has tried and I'll go with that," said the coach.
The other 14 ties are being played on Wednesday and Thursday, with the return legs due the week commencing November 8.
