Its report, which puts pressure on ministers who have so far been resisting a tax, also proposes a crackdown on marketing and advertising.
Food industry representatives say a new tax would be unfair on consumers.
The government will be setting out its plans early next year when it publishes a child obesity strategy, but has said a tax is not something it favours.
""The cross-party group of MPs acknowledged no single measure would provide a solution to the problem.
But the committee`s report said calls for a tax could "no longer be ignored".
It pointed to evidence from Mexico which introduced a tax on sugar-sweetened drinks of 10% and saw a 6% reduction in consumption.
The MPs urged the government to use the strategy to take strong action on the issue, pointing out that a fifth of children start primary school overweight or obese, rising to a third by the time they leave.
Committee chair Dr Sarah Wollaston said: "We believe that if the government fails to act, the problem will become far worse.
"A full package of measures is required and should be implemented as soon as possible."
The report from MPs comes just a month after a review by Public Health England, which also backed similar measures.
TV chef Jamie Oliver has also called on ministers to introduce a tax on fizzy drinks, saying it is the "single most important" change that could be made.
Sugary drinks are the single biggest source of sugar for 11 to 18-year-olds.
They get between 12 and 15% of their energy from sugar, but official recommendations say it should be less than 5%.
""But Food and Drink Federation director general Ian Wright said the committee`s report was "disappointing".
"No-one seems to have considered hard-pressed consumers in all this. Consumers already pay billions in VAT on food and drink," he said.
"As a result of the arbitrary new tax recommended by the committee, which, if introduced, would inevitably be increased year-on-year and extended to other foods, would leave consumers paying significantly more, every week, for the products they love."
Public Health Minister Jane Ellison said: "This government is committed to turning the tide on childhood obesity. That is why we are developing a comprehensive strategy looking at all the factors, including sugar consumption, that contribute to a child becoming overweight and obese. This will be published in the coming months."
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