Soares headed home for the tournament debutants in the 90th minute to cancel out Gabonese poster boy Pierre-Emerick Aubameyang's opener in the Group A game at the Stade de l'Amitie in capital Libreville.
Four-time winners Cameroon, who have been hit by a raft of withdrawals, and
2013 runners-up Burkina Faso were set to meet at the same venue at 1900 GMT knowing a victory would leave them on top of the section after the opening round of games.
After an opening ceremony in which French rapper Booba was the star turn, Gabonese President Ali Bongo declared the tournament officially open.
Bongo was accompanied by his counterpart from Guinea-Bissau, Jose Mario Vaz, and Issa Hayatou, the veteran head of the Confederation of African Football, at the 40,000-capacity stadium.
Those who wanted to see the host nation get off to a winning start in the 60th anniversary Cup of Nations were to be disappointed.
After a first half that failed to catch fire, Gabon came to life in the second half and opened the scoring in the 52nd minute.
The Guinea-Bissau defence failed to clear their lines and talented winger Denis Bouanga's low ball to the back post was converted by Borussia Dortmund star Aubameyang, to the delight of the home fans.
However, Gabon began to sit back and Guinea-Bissau had opportunities to equalise towards the end, with Rudinilson Silva and substitute Frederic Mendy both coming close.
They looked to have blown their chance, but they drew level in the final minute, Soares getting in front of Bruno Ecuele Manga to head home Zezinho's free-kick and allow the 'wild dogs' to claim a point from their first ever match at the Cup of Nations finals.
José Antonio Camacho (Head Coach, Gabon)
There were three points at stake but we lost two. However, we have the same chances to qualify as if we won. We knew the game was going to be tough. There is nothing easy in football. We have to take the matches one after the other, and they are all different. We have the next game against Burkina Faso, a team we have to study.
Baciro Candé (Head Coach, Guinea-Bissau)
We are very happy. Despite the difficulties in our country, we qualified for the finals. We knew that Gabon has an excellent team. We also know that they have the best player in Africa and Europe. We have our limits, but we have come here with ambitions. We will improve our skills over the course of the matches. We will try to go further. It is a very difficult competition. We only have a few days to rest. Cameroon is going to be difficult.