The Black Stars set up a semi-final date with Cameroon on Thursday following a 2-1 win over DR Congo at Stade d'Oyem on Sunday.
Andre Ayew scored the winner from the spot on 78 minutes after Paul-Jose Mpoku cancelled out Jordan Ayew's opener.
"Congratulations to the Black Stars for defeating the DRC to book a semi-final place at the Afcon," Bawumia said via his official Twitter account.
"Next up is Cameroon, which won't be an easy task.
"But with the continuous support and prayers of all Ghanaians, the Black Stars will make it.
"To the team, please stay focused. You can do it."
Ghana are seeking to win the Afcon for the first time in 35 years.
Sunday's victory, meanwhile, makes it six semi-finals in a row at the Afcon.
Their last outing in Equatorial Guinea two years ago ended in a shootout defeat to Cote d'Ivoire in the final.