The victory places Amavubi at the top of Group C with Nigeria who earlier edged Libya 1-0 on four points each.
Eyama was again called into action midway through the first half coming out of his line to punch away Eric Rutanga's left footed inswinger.
Equatorial Guinea's Pedro Oba missed from close range after Rwanda goalie Eric Ndayishimiye fumbled a set piece to path but he hastily fired over on the half hour mark.
Amavubi started the second half with some neat exchanges in midfield and through the wings.
Midfielder Djihad Bizimana forced a save from Eyama four minutes after the restart collecting a pass out of the area and cuts in to his right before smashing a low drive which the keeper parried to keep the score level.
Pedro Oba caused the Rwandan defence problems and could have put Equatorial Guinea in front but failed to connect to cross when he was unmarked at the back post on 64 minutes.
He was made to pay for his missed opportunity when Manzi glanced in Bizimana's corner from the left past the helpless Eyama to put the Amavubi in front.
Eyama pulled a magnificent acrobatic save on 83 minutes to deny substitute Muhadjiri Hakizimana from hitting Rwanda's second.
The defeat knocks out Equatorial Guinea after they went down heavily against Libya in their first outing whiles Rwanda, Nigeria and Libya all have a chance of making the last eight.
Antoine Hey (Head Coach, Rwanda)
We are very happy for this victory. I thank my players for this victory, after the draw against Nigeria. We did not have a good match. We were too hesitant. I feel that there is an improvement in the team’s performance. Equatorial Guinea created a lot of problems for us, especially on counter attacks. The result of the match Libya-Nigeria suits us. We will approach the match of Libya with the same determination.
Rodolfo Bodipo (Head Coach, Equatorial Guinea)
We tried to create some problems for the opponent. We have a young team and we are doing well. The players played without pressure. It was really cruel for us today because we played well, but we did not win. We made mistakes that we paid cash. We have gained a lot of experience. Our goal was to have experience. Even as if we are eliminated, we will play hard against Nigeria.