Mickael Pote had handed Benin a surprise lead after just 103 seconds, but Kwesi Appiah's side hit back through goals from Andre and Jordan Ayew before the interval.
Boye was dismissed after a controversial second booking in the 54th minute for apparent time-wasting, giving Benin the impetus to secure a point nine minutes later through Pote's second of the game.
Ghana face Cameroon in Group F on Saturday, while Benin face Guinea-Bissau on the same day.
Benin started in stunning fashion, taking the lead inside two minutes when Pote latched onto Cebio Soukou's throughball, cut inside Lumor Agbenyenu and poked past Richard Ofori.
Their joy was short-lived, however, as Andre Ayew unleashed a powerful drive inside Fabien Farnolle's left-hand post in the ninth minute to draw Ghana level.
Despite Benin looking more threatening in attack, it was Ghana who struck again before the interval. Kasim Adams found Jordan Ayew and the forward thundered a right-footed strike into the roof of Farnolle's net in the 42nd minute.
Yet any hope Ghana had of a routine second half were blown apart nine minutes after the restart when Boye received a yellow card from referee Youssef Essrayri after stepping over a free-kick to leave it for goalkeeper Ofori to take.
Benin took full advantage of their numerical superiority shortly after the hour when Pote flicked home Jodel Dossou's low cross from the left.
Only a superb save from Farnolle prevented Thomas Partey's 25-yard free-kick finding the back of the net, forcing Ghana to settle for a point from their opening outing at the tournament.
Key Opta Facts:
- This is only the second time that Benin have avoided defeat in an Africa Cup of Nations game (D2 L8).
- Ghana have lost only two of their last 14 group games in the tournament (W9 D3).
- Mickael Pote is the first Benin player to score a brace in an Africa Cup of Nations game.
- The Ayew brothers have scored seven of Ghana's last nine goals in the Africa Cup of Nations.