Max Gradel's stunning strike in Malabo was enough to book a place in the last eight for Renard's side, the 1-0 win over Cameroon ensuring they finished top of Group D.
The Ivory Coast failed to impress in their opening two games, and were still far from at their best against Cameroon, but one moment of magic was enough to keep them in the tournament.
Renard now expects them to go from strength to strength in the knockout stages, starting with their quarter-final clash with Algeria on Sunday.
"I said before this game that it's not necessary in a Cup of Nations to start too quickly because it's a very difficult competition and with players like this they are ready to compete and they love games like this," said the Frenchman
"For a team like this, the Africa Cup of Nations begins in the quarter-finals because to go out before is a failure.
"If you don't get through with a team like Cameroon or the Ivory Coast or Senegal you get the finger pointed at you.
"The Algeria game will be very difficult - another one after a really tough group - so there you go, the competition begins now."
Cameroon coach Volker Finke praised his players for giving their best to try and turn the game around but could not hide his disappointment as the four-time champions finished bottom of the group.
"I think really the Ivory Coast took advantage of a little mistake we made in the first half to score the goal," said Finke.
"We tried really hard. Everyone gave their best, so that's why we can't say the team didn't try to the end.
"In the second half everyone gave a lot but there weren't many clear chances. In the first half we had two or three really good chances but didn't score and that is the reason we lost."
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