Prevailing in the first Test in Brisbane 23-21, the Lions then lost the second Test 16-15 to Australia in Melbourne, but Gatland's side stormed to a 41-16 win in Sydney to clinch the series.
Their points total in Sydney was a record for a Lions side in any Test match while Leigh Halfpenny's total of 21 was also a record for any player in a Test.
And British & Irish Lions chief executive John Feehan paid tribute to the class of 2013, revealing his pride at their invitation to Downing Street.
"We are extremely proud of what the Lions achieved over the summer, becoming the first touring party to win a series for 16 years," said Feehan.
"And we're equally proud that our achievements have been recognised with an invitation by the Prime Minister to Downing Street."