But they remain eight points behind the Catalan side, who are yet to drop a point this season.
In injury time José Casado was dismissed for the home side after receiving his second yellow card.
The green light was finally given to play the game on Monday evening at Rayo's Estadio de Vallecas ground in what was a tough battle for Madrid against city rivals Rayo with the pressure on after Barcelona beat Granada 2-0 on Saturday, their fifth win in succession, while Madrid had only one.
"We played as a team but we could still have been better. We were committed, which is important and it allowed us to win without problems," said Mourinho afterwards.
"It has been a strange situation with the postponement and it has been complicated to prepare for this but it was the same for both teams. We overcame it and although it wasn't a great win, we did enough. We had a lot of chances to score while they had few."
Rayo coach Paco Jémez felt that his side did well but the quality of Madrid shone through in the end.
"Against top sides it isn't enough just to compete. I can't criticise the team though, quite the opposite, I congratulate them for their effort," he said.
"We had the game more or less under control and it was when we were at our best that we conceded the second."
Sergio Ramos returned to the side after being dropped for Madrid's Champions League victory over Manchester City last week, while Michael Essien was also given the nod over Sami Khedira to partner Xabi Alonso in front of the defence in a more offensive formation than against the Premiership side as Luka Modric returned.
Jémez has shown an admirable commitment to attacking football and they faced the champions with two wingers in Jose Carlos and Lass Bangoura, who had full licence to go forward.
Apart from a dangerous header from Pepe, saved by keeper Ruben Martinez, Rayo took the game to Madrid, who were reduced to committing a series of fouls and Essien was yellow carded for a handball.
But Madrid conjured up an excellent goal on the counter-attack which spanned the length of the pitch and was finished off by Benzema, who knocked the ball in from close range from a Angel Di Maria cross on the left.
Madrid were not comfortable at the back and Alvaro Arbeloa gifted the ball to Lass at the start of the second half but he failed to control with just Casillas to beat.
In an open game Essien and Ronaldo shot narrowly wide as did José Carlos for Rayo before Ronaldo's penalty decided the game after 70 minutes.
It was the Portuguese's cross that was handled by centre-half Jordi Amat and Ronaldo took the spot-kick, sending the keeper the wrong way.
Minutes later Ronaldo could have extended Madrid's advantage but he hit the post from a Gonzalo Higuain pass with the goal gaping.
In the second minute of injury time frustration got the better of Casado. who left his foot in as he challenged Mesut Ozil and was given his second yellow card. – Sapa