Edin Dzeko and Mohamed Salah were the culprits in front of goal for the visitors before two goals in four second-half minutes ensured Madrid's first win in a two-legged tie over Italian opposition in Europe for 29 years.
Ronaldo swept home Lucas Vazquez's cross 26 minutes from time before turning provider for Rodriguez to smash home the second.
"I am happy with the way we attacked and I think defensively we improved a lot in the second-half," said Real boss Zinedine Zidane.
Roma coach Luciano Spalletti criticised his side's mental strength after letting Madrid off the hook.
"We have to point out that we had more chances than them and if we'd taken them then the psychology of the game changes," said Spalletti.
"Right now on a mental level perhaps we are a bit fragile. It's a big game, big games make us nervous and we have to change this."
Gareth Bale started for Madrid for the first time in over six weeks due to a calf injury and the Welshman began brightly as he got a rare chance to start on the left with Ronaldo through the middle in the absence of the injured Karim Benzema.
However, it was Roma who enjoyed the best of the chances before half-time to get back into the tie.
Former Manchester City striker Dzeko failed to repay Spalletti's faith by handing him a start up front when he fired into the side-netting after Salah's pace tore through the Madrid defence.
Salah had a great chance himself but he too could only find the side-netting when clean through on Keylor Navas just before the half-hour mark.
Yet, the hosts could also easily have been ahead by the break as former Arsenal keeper Wojciech Szczesny was put to work in the Roma goal.
Ronaldo then had Madrid's best chance of the half when he was played in by Bale, but Szczesny again scurried from his line to block the three-time World Player of the Year's effort.
Szczesny was forced into action once more at the start of the second period to parry Rodriguez's volley.
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