The current director of BBC Vision will succeed Mark Thompson in the position after being chosen by the BBC Trust panel led by chairman Lord Patten.
He will be paid an annual salary of £450,000, less than the £671,000 earned by Thompson.
"I'm delighted that the Chairman and Trustees have decided I'm the right person for the job," Entwistle said.
"And I'm very excited about all that lies ahead. I love the BBC and it's a privilege to be asked to lead it into the next stage of its creative life."
Patten added: "George is a creative leader for a creative organisation.
"His experience of making and delivering great programmes that audiences love - built up through many years of working for the Corporation - will prove invaluable as he and his team work to ensure the BBC remains the greatest broadcaster in the world."
He continued: "Above all George is passionate about the BBC, is committed to its public service ethos and has a clear vision for how it can harness the creativity and commitment of its staff to continue to serve audiences in ever more innovative ways.
"I once again want to pay tribute to Mark Thompson. His creativity, vision and leadership have made him an outstanding Director-General of the BBC and he will be sorely missed."
Thompson will step down this autumn after eight years in the post.
"I think this is a brilliant appointment," said the outgoing director-general.
"George has shown himself to be an outstanding leader with an intuitive understanding of public sector broadcasting.
"He has a formidable track record as a programme maker and in recent years has also shown his calibre as a leader. I wish him and the BBC every success in the years to come."
Entwistle was appointed as director of BBC Vision in April 2011, having previously been the BBC's controller of knowledge commissioning and the controller of editorial standards for BBC Vision.
A philosophy and politics graduate of Durham University, Entwistle's previous jobs have included editor of Newsnight, deputy-editor of Tomorrow's World, executive editor of topical arts at BBC Two and BBC Four and acting controller of BBC Four.
Other candidates thought to have applied for the director general role included BBC chief operating officer Caroline Thomson and Ofcom boss Ed Richards.