The visitors remained defensively strong throughout to keep the tie finely poised heading into Tuesday night's return fixture at the Aviva Stadium.
Arter could not hide his disappointment at missing out on a priceless away goal, but was ultimately happy with the final outcome in Copenhagen.
"We could have done with scoring an away goal, but it's important not to go into the second tie losing," he told Sky Sports News. "Other than the first 20 minutes, I thought we kept them quiet.
"The World Cup feels a bit closer than it did before the game. Before the campaign if you offered us the chance to play Denmark in a one-off game, we would have bitten your hand off."
Ireland, who last qualified for the World Cup in 2002, are now unbeaten on their travels in more than two years.