A soft penalty in stoppage time allowed Equatorial Guinea to level when Tunisia seemed destined for the semi-finals after Ahmed Akaichi put the highly fancied North Africans 1-0 ahead in the 70th minute.
Balboa converted the kick to take the game to an additional 30 minutes and curled home a stunning free kick in the 102nd minute to win it and set off jubilant celebrations across the country.
Equatorial Guinea now take on either Ghana or Guinea, who clash in their quarter-final in Malabo on Sunday.
Saturday's match will be remembered as much for the fairytale win as for the violent scenes throughout the game and after the final whistle.
The Mauritian referee Seechurn Rajindraparsad was the central figure as he handed Equatorial Guinea a controversial lifeline and then found himself attacked by furious Tunisian players as he was sped from the field by security officials.
As the hosts partied, fist fights started between the rival players with both benches at one stage laying into each other in unseemly scenes.
Amazingly no players or officials were shown red cards.
"In my 15 years as a player and 30 as coach I've never seen
anything like it," Tunisia coach George Leekens told reporters.
"Today's result was forced. The referee made a huge error and we did not deserve that."
His counterpart Esteban Becker avoided questions about the
legitimacy of Equatorial Guinea's win.
"Refereeing decisions are a normal part of football," he said.
Tunisia were the better side in a game that had niggle almost from the start and made the breakthrough when Yassine Chuikahoui swept the ball out wide for Mohamed Ali Yakoubi to deliver a cross to Akaichi, who stole in behind the defence and finished.
As the game headed into five minutes of stoppage time, an innocuous looking challenge from Hamza Mathlouthi on Ivan Bolado saw the referee award a penalty that put Equatorial Guinea back in the contest.
That set off a massive sulk from Tunisia that continued through extra time, exacerbated when Aymen Abdennour gave away a free kick outside his area that Balboa expertly curled home.
It was the fourth goal of the tournament for the former Real Madrid winger, now the competition’s top scorer.
Earlier on Saturday at the Estadio de Bata, the Democratic Republic of Congo came from 2-0 down to beat Congo 4-2 to become the first team into the last four.
(Reporting by Mark Gleeson; Editing by Alan Baldwin/Patrick Johnston)
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