Striker Dorlan Pabon flicked a through ball with the outside of his right foot past a defender to Rodriguez, who sidestepped goalkeeper Diego Penny in the box and shot left-footed into the net.
The victory at the Estadio Nacional lifted Colombia into fifth place in the nine-nation South American group with seven points from four matches. They are three points behind leaders Argentina, who have played one more match.
Colombia, who have not reached the finals since 1998, appointed Argentine Pekerman after sacking Leonel Alvarez in November following a home defeat by Argentina.
Peru, whose coach Sergio Markarian was deprived of half a dozen players through injury, are one from bottom with three points from four matches.
The Peruvians had the better of the first half but were denied by some good goalkeeping from David Ospina.
"We lost to rivals of such class that they knew how to play for time and the referee (Argentine Nestor Pitana) gave them all the time they wanted," Uruguayan Markarian complained.
"The team did what they could and if they couldn't do more it was because we had too many limitations," he told reporters.
Peru have not beaten Colombia at home since the preliminaries for the 1982 World Cup, which was the last the Peruvians qualified for.
They face Uruguay in Montevideo next weekend when Colombia are away to Ecuador in Quito.