The $10,000 (£7,000) fund, set up by Ledger's ex-fiancee Michelle Williams and friends following his death in 2008, was given to Heathcote at the sixth annual Australians in Film Breakthrough Awards in Los Angeles on Thursday night (13May10).
She was hand-picked from a pool of applicants by The Dark Knight star's pals and colleagues, including Colin Farrell, who presented the honour.
Ledger's sister Kate and his mother, Sally Bell, were on hand to witness the prizegiving, and admit they are touched his memory can help others realise their dreams of acting.
Kate tells People.com, "It is very special. It's honouring our boy so we couldn't be prouder and we think it is an amazing scholarship that is going to give so many opportunities."
His mother added, "He had a huge passion for acting. That was the most important thing for him. It wasn't about the money. He had such a passion that just showed through regardless of how big the role was."
The inaugural scholarship, which includes flights to Los Angeles and introductions to agents and casting directors, was awarded last year (09) to Australian performer Oliver Ackland.
Ledger died from an accidental overdose in January 2008.
The inaugural Australians in Film Scholarship was set up by Ledger's ex-fiancee Michelle Williams, and is supported by the Ledger family, Nicole Kidman, Hugh Jackman and Baz Luhrmann.
Actress Rachel Griffiths and moviemaker Gregor Jordan fronted this year's (09) panel, which saw Australian performer Oliver Ackland pick up the $10,000 (£6,250) bursary in January.
And Law, who was one of three stars to replace Ledger in his film The Imaginarium of Dr Parnassus, is happy to have been given the chance to pay tribute to the late actor, who died last year (08).
He says, "I am delighted and honoured to do what I can to keep Heath's legacy alive. He was a brilliant actor and a wonderful man and it is only fitting that this scholarship has been developed in his name."
The winner of the next scholarship will be announced in February (10).