The one-time DC United youth coach has been in temporary charge of the Irn-Bru First Division outfit since former boss Gareth Evans resigned last month.
Burke, who was previously number two to Evans, will be assisted by former Celtic striker Mark Burchill, who will also sign on as a player once international clearance is granted following a spell with Thai club Esan United.
Livingston chief executive Ged Nixon hailed the appointments, saying: "Richie and Mark impressed us hugely during our recent recruitment process and we are absolutely delighted to have both 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 an 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 has all the attributes required to make a successful transition from playing into management. As his hometown club, we are delighted to give him an opportunity to start that journey.
"Most importantly, they both share our philosophy in how the game should be played, and from that perspective, they'll build on the strong foundations we already have in place at Livingston."
The new management team will start work immediately, with their first match in charge this Wednesday night against Hamilton.
Livingston sit third in the First Division, 18 points behind leaders Morton.