In a first half of few opportunities it was the hosts who looked the more likely to make the breakthrough.
Despite Madrid's dominance, their Czech visitors nearly took a surprise lead when Marek Bakos headed wide in the final minutes of the half.
Plzen were rejuvenated after the half-time whistle and began to cause Madrid more problems in front of goal.
Vaclav Prochazka was off target for the away side with a tame effort on the hour before Michal Duris could only fire wide.
However, Madrid snatched the victory in the first minute of injury time through Rodriguez's strike.