Captain Vincent Aboubakar scored both penalties, calmly slotting the first to the right side of the goal and the second to the left to complete a comeback that left home fans sighing with relief.
Cameroon had waited out a three-year delay to host this Africa Cup after being stripped of the 2019 tournament and then seeing their 2021 hosting delayed a year because of the coronavirus pandemic.
Burkina Faso were the first to score at the Olembe Stadium Yaounde with Gustavo Sangare netting in the 24th minute.
He scored after Burkina Faso had a header cleared off the line and hit the crossbar in the same move.
But Burkina Faso lost the lead with two pieces of defending at the end of the first half.
Bertrand Traore barged into Andre-Frank Zambo Anguissa to give Cameroon their first penalty, which was only confirmed after referee Mustapha Ghorbal of Algeria consulted VAR.
VAR is being used in all games at the Africa Cup for the first time. It was used from the quarter-finals onwards at the last Africa Cup.
Soon after, Issoufou Dayo mistimed a sliding tackle to foul Nouhou Tolo and Aboubakar scored from the spot again three minutes into first-half injury time.
Burkina Faso threatened a second-half comeback as Traore tested Cameroon goalkeeper Andre Onana with a free-kick from range and Adama Guira then had an attempt saved by Onana, but the home side held on.
The Africa Cup officially started with a short opening ceremony a few hours earlier at Olembe Stadium.
Coronavirus restrictions limited the crowd at the Olembe Stadium to 80 percent of its 60,000-capacity, but that did not spoil the party for the hosts and celebratory fireworks lit up the sky after the match.
The match and opening ceremony were attended by Cameroon’s 88-year-old President Paul Biya, who has ruled the Central African country for 40 years, as well as FIFA President Gianni Infantino and Confederation of African Football head Patrice Motsepe.
The Cape Verde Islands and Ethiopia are due to play in the second Group A game later on Sunday at the Olembe Stadium.