South Sydney Rabbitohs star Inglis touched down in the first half as the Kangaroos twice took the lead in a game they had entered as favourites in Brisbane.
However, the Kiwis kept them in check to go in level at 12-12 at the break, by which time Inglis' flu got the better of him and forced him off.
Fellow back Daly Cherry-Evans was also forced off with a suspected glute injury and with their plans thrown up in the air, Australia struggled in the second half, ultimately going down 30-12 on Saturday.
"It backfired - it didn't work out," head coach Sheens said of Inglis' selection.
"Sam [Thaiday] had it during the week. GI pulled up a little sick [on Saturday] but everyone seemed to be thinking he was alright. He was dizzy, dehydrated. It backfired on us.
"They worked us over. Possession went against us. It was disappointing. The guys are all disappointed. We have to win two games now.
"We knew we were in for a tough game. If we're going to put any pressure on anyone you can't turn that ball over. You need to go set for set in Test match footy."
World champions Australia must now beat England and Samoa in their next two games to stand a chance of reaching the Wellington final, but New Zealand have put themselves in pole position of reaching the showpiece on home soil.
"I thought it was a great performance," New Zealand coach Stephen Kearney said.
"It's a really encouraging start to a pretty long tournament. We've got Samoa next week and I know a few guys there will be looking to take a few scalps in our team."