"Senator Edward Kennedy will be mourned not just in America but in every continent," Brown said in a statement following his death at the age of 77.
"He is admired around the world as the senator of senators. He led the world in championing children's education and health care, and believed that every single child should have the chance to realise their potential to the full.
"Even facing illness and death he never stopped fighting for the causes which were his life's work," Brown added.
The prime minister said he was proud to count Kennedy as a friend and he recalled that Britain had recognised his service this year with the award of an honorary knighthood.
Kennedy played a major role in establishing peace in Northern Ireland, helping to smooth the negotiations which led to the 1998 Good Friday Agreement ended decades of strife in the British province.
Former premier Blair, who oversaw the Good Friday negotiations, praised Kennedy's "passionate commitment" to the peace process.
He said: "Senator Kennedy was a figure who inspired admiration, respect and devotion, not just in America but around the world.
"He was a true public servant committed to the values of fairness, justice and opportunity.
"I saw his focus and determination first hand in Northern Ireland where his passionate commitment was matched with a practical understanding of what needed to be done to bring about peace and to sustain it.
"My thoughts and prayers today are with all his family and friends as they reflect on the loss of a great and good man."
Kennedy, the brother of former President John F. Kennedy died after a long battle with a brain tumour.