Pierre Webo, whose extraordinary point-blank miss was chiefly responsible for the first leg ending goalless, scored early on this time and Benjamin Moukandjo added a second before the break.
The opener came just three minutes in when Webo dispossessed defender Karim Haggui and produced a neat low left-footed finish across keeper Moez Ben Cherifia and in off the far post
Cameroon dominated the early stages and Benoit Assou-Ekotto's long-range shot took a deflection and an awkward bounce in front of Ben Cherifia, but he dealt with the ball.
However, Moukandjo made it 2-0 in the 31st minute when he dribbled through three Tunisian defenders and fired low inside the near post. His exuberant celebrations brought him a yellow card but the goal took his side a big step closer to Brazil.
Samuel Eto'o failed to shape a curling effort inside the far post but Tunisia threatened when Syam Ben Youssouf headed wide from a dangerous set-piece.
And Akaichi cut the deficit when he raced past defender Aurelien Chedjou on to a bouncing through-ball and lashed a first-time left-footed shot past keeper Charles Itandje.
He almost made an unwanted contribution at the other end as Makoun's header from a corner deflected off him and just over the crossbar.
But another set-piece took Cameroon clear as Assou-Ekotto's corner was headed home at the near post by the unmarked Makoun.
Moukandjo missed a glorious chance to kill the tie off once and for all when, one on one with Ben Cherifia, he first side-footed straight at the keeper and then casually skied the rebound over the bar.
The same player's clumsy touch ended a classy Cameroon break from a Tunisian corner and Makoun failed to play in Eric Choupo-Moting.
But Makoun netted his second, and Cameroon's fourth, when he tapped into an empty net after Choupo-Moting's low shot came back off the post.