A dull first half passed largely without incident, but Tunisia went ahead through Ahmed Akaichi in the 70th minute.
Tunisia looked destined for a place in the semi-finals until Hamza Mathlouthi was controversially deemed to have fouled Ivan Bolado in the penalty area deep into second-half stoppage time.
Having looked unlikely to find the net for much of the encounter, Equatorial Guinea drew level when Balboa held his nerve to slot home from the spot.
And Balboa delivered the knockout blow 12 minutes into the additional 30 - finding the top right-hand corner with a stunning 25-yard free-kick to send the majority of the Estadio de Bata crowd into raptures.
The hosts clung on for an unlikely victory and will now meet the winner of Sunday's clash between Ghana and Guinea in the last four.
Equatorial Guinea, who brought Diosdado Mbele in for Dani Evuy, were put on the back foot in the opening stages, with Yassine Chikhaoui coming agonisingly close to getting on the end of a Wahbi Khazri cross.
An unchanged Tunisia side continued to appear the more likely to break the deadlock, and Aymen Abdennour almost pounced to punish goalkeeper Felipe Ovono when he spilled the ball from a corner.
Ovono recovered to prevent Equatorial Guinea from going behind, but Tunisia were beginning to find holes in the hosts' defence.
A neat attacking move was halted by an offside flag, and Ferjani Sassi should have done better when he sent a diving header wide of the right-hand post from an Akaichi cross shortly after the half-hour mark.
Equatorial Guinea finally showed signs of life as the match approached half-time – Balboa seeing an effort hacked away by Abdennour after Aymen Mathlouthi opted to punch a cross but failed to alleviate the pressure.
Akaichi forced Ovono to tip over his crossbar with a venomous strike from the edge of the box six minutes into the second half, before Khazri sent a long-range strike just wide.
Mohamed Yacoubi conceded a free-kick in a dangerous position when he hacked down Balboa in the 63rd minute, but Equatorial Guinea were unable to make the most of a rare chance to test Aymen Mathlouthi.
Ovono kept out a Hamza Mathlouthi strike from distance soon after, but was left writhing around in apparent agony when Akaichi charged in to try and convert the follow-up.
Tunisia eventually broke the deadlock in the 70th minute when Akaichi turned the ball in from a superb Hamza Mathlouthi cross.
Aside from Emilio Nsue's late tester for Aymen Mathlouthi, Equatorial Guinea struggled in their attempts to find a leveller, but Hamza Mathlouthi's alleged foul on substitute Bolado in stoppage time presented Balboa with the chance to force extra time from the spot.
Bolado hit the deck with minimal contact, prompting a furious response from the Tunisia players, but Balboa made no mistake with his spot-kick, which was driven low to the left of Aymen Mathlouthi.
And Balboa made himself even more of a hero with supporters in his homeland in the 112th minute with a sublime free-kick to settle the encounter, although Ovono had to be on his guard to keep out an Ali Maaloul set-piece and a powerful Sassi header before time was up.
Youssef Msakni wasted Tunisia’s best chance to send the game to penalties, though as he poked the ball wide from close range and the hosts held on to reach the last four.
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