"I did not mean for it to go like this," he was filmed saying in one video, to laughs from the audience gathered on the footpath.
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Pierce, who previously starred as Robert Zane in Suits, thanked people for evacuating the building safely and apologised on behalf of the show's cast.
"We are so honoured that you came tonight; we're so sorry that ... unfortunately this happened.
"We promise you when we return we will make this a memorable evening at the theatre, no matter what," he continued in another video.
"So as of now, turn to your loved ones and friends, to your left and right, and say 'Which pub are we going to?' and let's go there and have a memorable night in anticipation of coming back."
Samuel Rhydderch, 21, who was at Wednesday night's show, said: "There was a piercing scream coming from the upper tier above us, then there was a lot of rumbling which turned out to be people running. I saw some people being treated for cuts and bruises but it didn't seem serious."
- additional reporting nzherald.co.nz