James Rodriguez opened the scoring for the South Americans in the 64th minute of the Group C clash, before Juan Quintero doubled the advantage six minutes later.
Gervinho found the net to put Colombia under some late pressure, but Pekerman's men held on for the win.
The three points put Colombia on the brink of qualification for the knockout stages, and Pekerman was full of praise for his squad.
"We are learning to be in a World Cup," he said. "I'm very happy for the performance of everyone."
One man who will have particularly delighted Pekerman is Rodriguez, whose second-half strike sparked the game into life.
"(Ivory Coast were) a tough opponent who has big players," said the forward. "I think we all have to be happy.
"First I am happy because we won against a tough rival with great players, but the only thing I want is to help the team win and, if I can score a goal, it is even better."
‘Even though it was late in the game, the Arsenal fans started singing my name. It felt like the moment I announced myself as a professional footballer’: Ex-Gunners star recalls scoring first senior goal as a teenager at Highbury
'I started to drive myself crazy': Former Manchester United midfielder explains why it all went wrong for him under Erik ten Hag
