The Danes inexplicably missed the top mark, sailing well beyond the mark before realising their error and turning back. It cost them seven places, with the Danish pair eventually finishing eighth.
Australian defending champions Nathan Outteridge and Iain Jensen - considered the New Zealanders' chief rivals - had an expensive day, recording 13th and eighth placings to sit in 11th overall.
Burling, in his typical understated style, was not getting carried away with his team's opening performance but said he was pleased with the consistency shown.
"We're really happy to get a couple of low scores on the board and not walk away with anything big and just to get into the series," he said. "We've been waiting a long time to get racing, and we've been really enjoying ourselves out there and I think that showed today." Maloney and Meech were also happy with their opening gambit of fifth and sixth, which places them sixth overall but well in touch of the leaders. Just four points cover the first seven crews.
"We had a pretty consistent day," Meech said. "On each beat we were gaining a boat so that was quite nice and getting a feel for it. I think we shook a few of the nerves off a bit today."
Meech's brother Sam, sailing in the Laser class, also had a solid day. He improved one place to sixth overall following 13th and sixth placings but, with just two races remaining in the Laser fleet before Tuesday's medal race, will need a strong day today to put himself in medal contention. He is 17 points behind Croatian leader Tonci Stipanovic.
Only one race was possible in the women's 470 class but it was one to forget for defending champions Jo Aleh and Polly Powrie.
They lost the overall lead following a 12th placing, dropping to sixth at the midway point of their 10-race qualifying series. British winners and 2012 silver medallists Hannah Mills and Saskia Clark assumed the lead but are only six points clear of the Kiwis on a congested leaderboard.
Team Jolly's disqualification in the opening race of the regatta on Thursday is putting the pair under extra pressure. Had their original sixth place finish stood, Aleh and Powrie would have been able to discard today's results and would have been joint leaders with the Brits.
It was a disappointing day for the men's 470 crew as well, after Paul Snow-Hansen and Dan Willcox slipped from 11th to 13th overall when they limped home 23rd in the only race possible.
Both 470 fleets will have a reserve day tomorrow, while the fourth-placed Nacra 17 pair of Gemma Jones and Jason Saunders, along with 15th-placed Finn sailor Josh Junior, will be back in action after a break in racing today.