The petition.parliament.uk site is designed to help people get issues on the political agenda and open up a "new dialogue" with MPs and ministers.
E-petitions will initially be considered by a cross-party committee of MPs chaired by Labour's Helen Jones.
The first e-petition tabled called for increased taxes on foreigners buying UK property worth more than Â£3.5m.
Under the previous system, petitions were directed to government departments rather than MPs, with 10,000 signatures required for an official response and 100,000 needed for a petition to be considered for a debate in Parliament.
More than 40 petitions passed the 100,000 signature threshold during the last Parliament. A number of them were subsequently debated in Parliament although critics say not enough of them were given parliamentary airtime.
The thresholds for a petition getting a response or being considered for a debate have not changed in the revamp of the old website which launched in 2011. The new site will be jointly run by the House of Commons and the government.
The Petitions Committee, made up of 11 MPs from different parties, said it would enable every British citizen and UK resident to start or a sign a petition electronically.
The committee, whose creation was recommended in a report by MPs last December, said its role was to liaise with petitioners, seek information from government about issues raised and, in some cases, refer petitions to specialist parliamentary committees.
"First and foremost, it is an easy way for people to let MPs and ministers know they care about," Ms Jones said.
"What's more, it gives us a chance to open up a new dialogue between the public and Parliament. I very much hope that the petitions committee will be able to bring forward issues that wouldn't otherwise have been debated or investigated in Parliament.
"For the most pressing petitions, we will be able to schedule debates...It is an exciting step for Parliament and a visible demonstration of its commitment to improving public engagement with what goes on here."
Deputy Commons leader Therese Coffey said the new website would further improve public engagement with politics.

Omar Khan, 31, had worked at The Johnson Partnership in Nottingham for five years before he was arrested.
Erlin Manahasa, Albert Dibra and Nazaquat Ali joined Khan in admitting the same charge, between 1 October  and 4 December last year, at Nottingham Crown Court.
They are due to be sentenced on 15 April.
Updates on this story and more from Nottinghamshire
The court heard the case involved the recovery of 1kg (2.2lb) of cocaine.
Digby Johnson, a partner at the Johnson firm, confirmed they did not represent Khan - who had set up his own office and was set to leave the company.
"I still find it hard to believe he could do something as stupid as involve himself in drugs and people who were heavily involved in drugs," Mr Johnson said.
"I'm just at a loss. You do question everything you've ever believed about anybody."
Mr Johnson also described Khan, of William Street, Huddersfield, as a "lovely guy" who was "incredibly hard-working".
"He would put in hour after hour," he added. "He was keen to progress and impress."
Khan appeared at the court via video link to admit conspiracy to supply the Class A drug.
Manahasa, of no fixed address, Dibra, of Joyce Avenue, Nottingham, and Ali, of Chard Street, Nottingham, were remanded in custody following the hearing.

Open-side flanker Sam Underhill has been ruled out for four months with a shoulder injury.
McCusker made 133 appearances for Scarlets before moving to London Irish last season, where he played 17 games.
The 30-year-old has won 10 caps and joins Ospreys having initially signed for Welsh Premiership club Carmarthen Quins following his release from Irish.
"With Sam's injury and one or two others working their way back to full fitness who are not quite there yet, we felt that we needed someone else in to help us through this period," said Ospreys head coach Steve Tandy.
"We are fortunate to have someone with Rob's versatility and experience available and have brought him into the environment with a view to him helping us prepare for Zebre at home in the opening round of the Pro12 next month."
Members of the public can once again digitally petition MPs on issues with the launch of a new website.
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A barrister who was due to move into his own chambers in Huddersfield has pleaded guilty to supplying cocaine.
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Ospreys have signed former Scarlets and Wales back-rower Rob McCusker on a short-term contract as injury cover.