Former Celtic and Scotland striker Mark Burchill - who will continue to be registered as a player - will be his assistant and both have signed contracts until the summer of 2015.
Burke, the former head of youth development at the club, was made caretaker following the departures of John Collins - who resigned at the end of February - and his assistant Gareth Evans, who was sacked.
Chief executive Ged Nixon said: "Richie and Mark hugely impressed us in our recent recruitment process and we are absolutely delighted to have them on board.
"We see them as the ideal candidates to take forward our hopes and aspirations for the future - to develop our own quality young players and deliver and attractive and successful team at first-team level.
"Richie has worked closely with the club since last summer when he arrived to take up a post with the Livingston Youth Foundation and he has extensive coaching experience from his time in the USA.
"Mark is at an earlier stage in his coaching career but he has all the attributes required to make a successful transition from playing into management."