Labour was in power in 2001 when the Australian government removed most entitlements from New Zealanders who have moved across the Tasman since then.
Mr Key said there would always be some issues with Australia, but New Zealand got a lot out of the relationship.
"So many things go well and go right," he told TVNZ's Breakfast.
"There's a million things that we do with Australia and, like any relationship that is that tight and that broad and deep, there will always be one or two issues."
Among the current issues is Australian supermarket chains Coles and Woolworths dropping New Zealand-made products, even if an alternative cannot be sourced locally.
Mr Abbott has refused to intervene but has encouraged New Zealand companies complain to the Australian Commerce Commission.
Mr Key said there was nothing new about supermarkets running a campaign, and Australian consumers were the ones who would miss out.
"In the end, it's the Australian consumer that will eventually say, 'You know what, I actually want to have a New Zealand product because it's better.'"
Mr Key said there was very little Mr Abbott could do.
"But what the Australian government has done is set up a Government review of competition in effectively supermarkets in Australia."
Mr Cunliffe has said the shut-out was a contravention of the transtasman Closer Economic Relations (CER) agreement.
But Mr Key said Crown Law had taken the view that it was not in breach of the CER.
"Great if it is, because if it is, we can take the case. But that doesn't appear to be the case."