Tauranga's chances became few and far between, but in a rare attacking move, the ball fell to Kenny whose deflected shot found its way into the net. Moments later, Birkenhead were down to 10 players when Tom Davis was sent off and it seemed momentum would swing the home side's way.
But it was lively Birkenhead substitute Godwin Darkwa who rifled a low shot to Pamment's right to score the winner.
"They couldn't have scored at a better time than just after they went down to 10 players," said Tauranga head coach Duncan Lowry.
"I have no complaints about it because I think we got out of jail. I thought we were lucky to keep it to one [up to that point], and yet we could have pinched it.
"Saying that, everything in this league is going to be a learning curve for everybody at this club and there's no easy beats in this league."
Injured Jordan Culpepper did not make Sunday's squad.
"We maybe missed Jordan Culpepper a bit - he hurt himself and he came off last week. Jordan's an honest player, he's a tough as they come but he wasn't 100 per cent fit, so rather than have someone who isn't 100 per cent fit on the bench, I'd rather rest him for the week and get him ready for next week.
"I thought Andrew Cooper gave us a bit more on the left and Mark van de Salm showed up well."
Hughes and skipper Raymond How are new injury concerns after Sunday's game.
Goalkeeper Tom Pamment had a top first half.
"He did very well. The three or four saves he made when the ball was ricocheting around, he smothered the ball so you've got to count them as saves," said Lowry.
Next Saturday Tauranga City United again make the trip to Auckland, this time to take on Western Springs.