A claim that the drink gave a "great start to the day" suggested it promoted health, the watchdog found.
Nestle, the firm behind the drink, said it was "disappointed with the ruling" but that it would remove the strapline.
It added that it was "actively looking for solutions to help us reduce sugar."
An advert on Asda's own-brand milk labels for Nesquik hot chocolate featured a rabbit a cartoon character with the strapline "For a great start to the day!" and a badge labelled "Nutristart".
Following a complaint from The Children's Food Campaign, the Advertising Standards Authority (ASA) found that the ad in context suggested health benefits without putting forward specific health claims, in contravention of European law.
Nestle acknowledged that a 200ml drink made with three teaspoons of Nesquik hot chocolate contained 20.2g of sugars, which is "high" under the traffic light system of food labelling.
But it said more than half the sugar in the prepared drink came from lactose, which occurs in milk. It said the drink had added vitamin C and D, and zinc and iron.
A Nestle UK spokesperson said: "The advert for Nesquik Hot Chocolate shown on the label of a family-sized bottle of milk was undoubtedly targeted at adults who were shopping for their family, making it clear that the product should be consumed over a number of days, rather than in excess.
"However, we always listen to concerns when they are raised.
"As a responsible manufacturer and to remove any ambiguity in future, we will no longer use the statement: 'For a great start to the day!' in our UK advertisements."
Asda said it had no part in creating or approving the ad, which had been given directly to its milk supplier by Nestle.
An Asda spokeswoman said: "We respect the ASA's decision and will work with our suppliers to ensure the ruling is adhered to in future marketing."
The Children's Food Campaign said the "Nesquik bunny" was in a "fresh marketing stew".
"It is the second time in almost as many years that we have forced Nesquik to change their advertising because it encouraged poor nutritional habits in children and could be seen to mislead parents about the health benefits of such a sugary product," said Malcolm Clark, coordinator of the Children's Food Campaign.
An ad for Nesquik hot chocolate featuring a rabbit cartoon has been banned by the UK's advertising watchdog, which found it encouraged poor nutritional habits in children.