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.
Members of the public can once again digitally petition MPs on issues with the launch of a new website.