The winger's first-time finish arrived before the 15-minute mark at Kansas City Stadium in the round of 32 tie.
Colombia dominated for long spells and finished the match with 20 attempts on goal to Ghana's eight, eight of them on target compared with none for their opponents.
Néstor Lorenzo's side missed further chances through Puerta and Luis Díaz in the first half, with Ghana goalkeeper Ati Zigi producing a strong save to deny Daniel Mojica.
Díaz thought he had doubled Colombia's lead with two opportunities just before the hour mark, but a tight offside call and a missed one-on-one with the goalkeeper kept the score at 1-0.
Ghana failed to register a single shot across the full 90 minutes.
Colombia will now face Switzerland in the last 16.
Ghana head coach Carlos Queiroz conceded his side had been second best, despite praising his players' effort.
"I think generally speaking the Colombia team controlled much better the game — the passing, the movement — and doing that they damaged our fitness and our ability to recover the ball and try to go forward," he said.
Queiroz pointed to moments when Ghana pushed forward with intent, even if they could not turn possession into clear chances.
"But there was some moments when the Ghanaian team with heart and with full determination we put pressure on them, we tried to go forward, we tried to create opportunities," he said.
The head coach identified quality in the final third as the decisive factor in the game.
"But for any reason, the last pass, the last service was not with the quality that was necessary in this game," Queiroz added.
He was in no doubt about the outcome once the full 90 minutes had been played.
"So generally speaking, I think the best team won the game," he said.
The result ends Ghana's World Cup campaign at the hands of South American opposition, a pattern that has repeated itself for the Black Stars at previous tournaments.
Colombia, by contrast, topped Group K ahead of Portugal and go into the Switzerland tie unbeaten, having conceded just once in the tournament so far.
Attention at the Colombia camp now turns quickly to preparations for the last-16 clash with the Swiss.