Lock Matt Thompson went over beside the posts and then from a crisp 10m lineout play Ah Chookoon powered his way through three weak attempts to score down the blindside.
Pirates kept it up with tries either side of halftime.
First, hooker Junior Tume was successful at the third attempt for a dive at the line after Ulukuta and halfback Ricky Boniface had a crack, then centre Samu Eteuati took off on a 40m dash with the ball in one hand, embarrassing the Marist back three.
Holding a 26-point buffer with the bonus point in the bag, Pirates got a little cocky finding the front of their own goal posts a fair place to spread the ball.
Out-matched, Marist nonetheless never stopped trying as first-five Marek Willis finally cut through a gap to score under the posts.
The sting seemed to jolt Pirates back into focus as they took advantage of an out-on-the-full Marist clearance with second-five Soo Mulu selling everyone a subtle dummy to dot down beside the posts.
Still looking to spread in their own half, a Pirates fumble saw Marist No 8 Bradley Graham prove his soccer skills as he kept the ball on his toe and won the race to the in-goal.
For Marist coach Greg Proctor, missing five regulars in key spots, like halfback and the front row, meant he was treating the result as an aberration.
"We always knew we were going to have a hard game.
"We just didn't get enough ball, we were defending for long periods of time, but I couldn't ask for much more from these guys."
His opposite, Philip Morris, conceded being expansive in their own 50m had given Marist the better of the second stanza, but considering Pirates' advantage there was no harm in allowing it.
"These boys, it's organised chaos and, hopefully, going back to structure.
"They've got a licence to express themselves.
"There's enough old heads that we realise if we have to close it down, they can close it down."
Integrity Motors Pirates 31 (Samu Eteuati, Soo Mulu, Sanele Ah Chookoon, Junior Tume, Matt Thompson tries, Denning Tyrell 3 cons Dave Hoskin Carriers Marist 12 (Marek Willis, Blake Eves tries, Scott Dowman con). HT: 19-0.