The 47-year-old won promotion from the National League with Newport County in 2013. He left the club in 2015 to become manager of League One club Gillingham.
In January this year he left the Gills and was appointed manager at Northampton Town but after a difficult start to the season he left the club in August.
Edinburgh takes over from Steve Davis with Orient sitting 20th in the National League and without a win in 14 games.
Speaking on the appointment, Director of Football Martin Ling said: “We are extremely pleased to be able to welcome Justin to this Football Club. We built up a criteria for a Head Coach and Justin ticked all of the boxes. He has experience in the National League which I feel is of huge importance, but also enjoyed success in this division by achieving promotion with Newport.
“What I really liked about Justin’s CV was that he started in non-league and earned his stripes by succeeding at this level before building on that in the Football League.
“We are focussed on moving this Club forward, but the first hurdle is the most difficult in getting out of this division and that has been proven over the years with several teams that have dropped out of the Football League. Justin has overcome that first hurdle before and is someone that we know can be very productive. He has a record of pulling teams away from trouble and improving them during his tenure.
“It has been an extensive search and we said that we would leave no stone unturned. We feel as a Club that we have got the right man and the sooner Justin starts working with our players and gets this season turning around, the better for every single person that is connected to Leyton Orient.”
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