After a quiet start, Mosa Lebusa opened the scoring for the hosts in the 11th minute after a great leap inside the box to power through a header into the bottom corner.
Thembinkosi Lorch almost found the target for the Buccaneers in the 19th minute after hitting the return pass from Luvuyo Memela, but he ruffled the side netting.
Muhsin Ertugral made a tactical substitution on the half-hour mark, taking off striker Tashreeq Morris for Ndiviwe Mdabuka.
Yannick Zakri made it 2-0 almost immediately thereafter with a low finish inside the box after some good work by fellow new signing Tendai Ndoro on the right flank.
Pirates coach Milutin Sredojevic was forced into a change in the 35th minute after midfielder Mpho Makola got injured. He was replaced by Happy Jele.
The Urban Warriors suffered a forced change themselves five minutes before the break as goalkeeper Brandon Petersen could not continue due to injury and was replaced by Jody February.
The home side endured another blow as Isaac Nhlapo was introduced in the beginning of the second half, replacing injured goalscorer Lebusa.
Lorch had an opportunity to pull a goal back seven minutes into the second half, but his strike was smothered by February at the near post.
Innocent Maela missed a sitter minutes later after a glorious chance at the far post saw him send the ball over the bar at close range with the net wide open.
Lyle Foster was thrown into the fray with 20 minutes remaining on the clock in place of Maela, who had a couple of glaring misses upfront.
Sredojevic withdrew another fullback in Mthokozisi Dube with 12 minutes left for Thamsanqa Gabuza.
However, Zakri sealed the match in the 82nd minute by turning home a cut-back from debutant Siyanda Zwane on the right wing.
Ajax: Petersen (February 43’), Zwane, Booysen, Pieterse, Lebusa (Nhlapo 46'), Malepe, Morris (Mdabuka 31’), Margeman, Sasman, Zakari, Ndoro.
Subs: February, Lakay, Thabotha, Mdabuka, Julies, Nhlapo, Kammies.
Pirates: Mabokgwane, Munetsi, Nyauza, Dube (Gabuza 79'), Maela, Nyatama, Makola (Jele 36'), Memela (Foster 71'), Lorch, Kutumela, Shonga.
Subs: Mpontshane, Jele, Maphanga, Matlaba, Mlambo, Foster, Gabuza.