Rules which embed centuries-old discrimination against women in the line of royal succession belong to a "bygone era", Nick Clegg has said.
Ministers want to end the principle of sons taking precedence over daughters as heirs to the throne.
They also want to end rules barring the sovereign and prospective heirs from marrying a Catholic.
But some MPs raised concerns about speeding the legislation through Parliament without proper scrutiny.
Mr Clegg insisted the current rules had "no place in modern Britain".
Labour said it strongly supported the Succession To The Crown Bill, but warned of possible "unintended consequences".
Under current laws, dating back to the 1701 Act of Settlement, women are superseded by their brothers in succession even if they are the first born.
'Superiority of men'
Deputy Prime Minister Nick Clegg told MPs: "The current rules of succession belong to a bygone era. They reflect old prejudices and old fears.
"Today we don't support laws which discriminate on either religious or gender grounds. They have no place in modern Britain and certainly not in our monarchy - an institution central to our constitution, to the commonwealth and to our national identity too.
"With the Duke and Duchess of Cambridge expecting a baby and having just celebrated our Queen's 60 year reign, this bill is timely as well as popular. It is also straight forward and enjoys support across these benches."
The current rules were based on the "supposed superiority of men", Mr Clegg said. "That anachronism is out of step with our society. It sends the wrong message to the rest of the world and it's time for the rules to change."
The changes would apply retrospectively so that a child born after 28 October 2011 would be subject to the new rules, removing any gender preference from the laws of succession.
Mr Clegg revealed that final consent to the bill from the 15 other Commonwealth realms was received just hours before the Duke of Cambridge - who is second in line to the throne - and his wife Catherine announced they were expecting a baby, which is due in July.
As well as changes to succession, the bill will also remove a requirement for descendants of George II to seek permission to marry from the monarch, dating back to 1772, replacing it with a requirement for the first six people in the succession to seek the sovereign's consent.
The current prohibition on the monarch being a Catholic will remain in force, but members of the royal family who marry a Catholic will no longer lose their place in line to the throne.
'Royal Pandora's box'
The deputy prime minister said the "narrow" bill, with only five clauses, could be passed through Parliament quickly, but a number of MPs expressed concerns about allowing enough time to debate the constitutional change.
Conservative MP Jacob Rees Mogg, a Roman Catholic, argued it was an "insult to the nation and to our sovereign and indeed to Parliament" and said he was unhappy that any child of an heir to throne who was married to a Catholic would not be brought up in the religion.
DUP MP Ian Paisley said it could open up a "royal Pandora's box" and Labour's Paul Flynn said the bill's failure to address the prohibition of a Catholic monarch could actually "strengthen the prejudices of the past".
Conservative MP Nicholas Soames said that he backed the scrapping of the primogeniture rule but warned that the bill would "tinker away" with some of Britain's fundamental constitutional foundations.
"This bill touches upon British history and tradition and it succumbs to the rather passing enthusiasms of the 21st century. Above all the proposals interfere with statutes that have slept for over 300 years and a common law rule of far greater antiquity," he said.
Labour's shadow minister Wayne David said: "This bill is a small bill, which is significant however in terms of its constitutional implications.
"It reinforces and extends a process of modernisation for our constitutional monarchy, which has been underway for some time.
"The people of this country are quite rightly very supportive of our royal family and recognise that not only is the monarchy an important part of our nation's heritage it is also a vital element in defining the identity of Britain in the 21st century. These changes, in this bill, will help it to ensure the monarchy will continue to be an essential part of Britain's future."
According to recent media reports, the Prince of Wales - while backing the end to male primogeniture (the principle that males take precedence in the royal succession) - has expressed concerns to officials about "unintended consequences" which could affect the monarch's role as Supreme Governor of the Church of England.
A spokesman for the Prince of Wales has said any change to the law is a matter for the government.
Dismissing some media reports to the contrary, Conservative MP Sir Tony Baldry, who represents the Church of England in the Commons, said its leadership had no objection to the bill.
The proposed changes also need to be approved by the 15 other countries of which Queen Elizabeth II is head of state. They agreed in principle to do so in 2011 and to work to enact the necessary legislation simultaneously.
Republic, which campaigns for an elected head of state, called for "real democratic reform" in the shape of a referendum on the UK's future constitutional arrangements.
"It is absurd that in 2013 our parliament is debating a new law that would continue a hereditary succession, that would require people to ask permission to marry and ban Catholics from being head of state," its spokesman Graham Smith said.
"What we need is an end to the hereditary succession altogether. If we must wait for that to happen then the very least we could expect is an end to the ban on Catholics and a more sensible set of reforms."