Both teams had early efforts on goal, but it was Ghana who took the lead with a superb strike in the 11th minute.
Chile were back on level terms in the 23rd minute as Nicolas Castillo thundered home from a tight angle, before Angelo Henriquez put the South Americans in front with just under half an hour on the clock.
Ghana started the second half in positive fashion, but Eric Antwi was still called upon to deny Castillo and Henriquez.
The African team piled the pressure on their opponents and it finally told in the 72nd minute, with Clifford Aboagye setting up Assifuah to poke the ball beyond Melo.
The teams could not be separated after 90 enthralling minutes, prompting extra time. There was no let-up in the action, however, as the match continued in similar end-to-end fashion.
It was Chile who drew first blood, with Claudio Baeza lifting the ball over Antwi for Henriquez to grab his second of the evening.
Once again, Ghana threw caution to the wind and launched everything at the South American side. They drew level when Salifu calmly slotted past Melo in the 113th minute and penalties looked destined to decide the game.
However, as the contest entered the 120th minute, Acheampong got himself to the byline to cross for Assifuah at the back post, and he headed Ghana into the semi-finals.