Australian Johnson took over on an interim basis after Andy Robinson resigned in November and led the Scots to third place in the Six Nations, their best finish for seven years.
Johnson will remain in charge for June's quadrangular tournament in South Africa also involving the hosts, Samoa and Italy, where he will be joined by new forwards coach Jonathan Humphreys.
"Scott and I are working together on this and have thought long and hard about the best solution and people for Scottish rugby to continue to head in the right direction," Scottish Rugby Union chief executive Mark Dodson said in a statement.
"Johnno has been a popular coach with the players. He has said he enjoys hands-on coaching and his reputation in that role has been enhanced by his work with the Scotland team.
"But he also has other qualities in abundance that we wish to utilise and it was a decision that was made by both parties to get the best structures and systems in place to take Scottish rugby forward.
"The Director of Rugby is a hugely important appointment, one that is absolutely integral to Scotland's standing on the world stage in the next decade and beyond.
"Scott will lead on player acquisition and contracting. He'll be responsible for driving a new Academy structure and for heading up our coach development, which includes our priority of nurturing the next generation of Scottish coaches who can operate at elite level."
Johnson, formerly assistant coach of Wales and Australia, said: "The job excites me. It's about making a really positive contribution that will stand Scottish rugby in good stead for years to come and I am looking forward to the challenge of the new role as well as continuing with the Scotland team."
- Sports & Recreation
- Scottish rugby