Leader Ed Miliband also pledged to increase statutory paternity pay to £260 a week - in line with the minimum wage.
The move would cost the taxpayer at least £150m a year but Labour says this would be more than offset by savings in tax credits from extending free childcare.
He said: "At the heart of Labour's plan is the belief that Britain succeeds when modern working families succeed.
"That means giving dads, as well as mums, the chance to spend more time at home in those crucial weeks after babies have been born."
Labour said adopting the reforms, first put forward by the left-leaning IPPR think tank last year, would benefit up to 400,000 families a year.
Under existing rules, new fathers qualify for a statutory £138.18 a week, equivalent to £3.45 an hour for a 40-hour week, with employers encouraged to make up the gap in the employee's usual pay.
At the moment, just over half of new fathers (55 per cent) take it up.