"We lifted, but this time we had chances but couldn't put them in the back of the net," said King.
"It's a long way to go, not to get a result."
Ben Foxall got a double for the home side, with Ethan Dent getting their other goal.
Athletic's Quinn Mailman picked up a yellow card, as did Havelock North's Bjorn Christensen.
"The biggest thing, for me, was was seeing the guys bounce back from their performance last week," said King.
"We just gave ourselves too much to do."
Ryan Holden moved to the back to cover the injured Kelvin Hove and was named man of the match.
Athletic will now be very interested spectators as they take the bye week while Havelock North will finally play New Plymouth Rangers up at Merrilands in the top of the table clash for the second round, after their April 8 game was postponed due to rough weather.
Athletic have been the measuring stick as the one side to give both those teams a tough challenge, and King would like his players primed for when Rangers return to Wembley Park in just under two weeks.
"We know we can beat Havelock, that's the worst part, and we know we can beat Rangers."