Johnson replaces Bath captain Danny Grewcock in the second row and, as expected, reclaims the captaincy from fly-half Jonny Wilkinson.
Robinson, meanwhile, will be on the wing - his first start there in an England jersey for 17 months - with Josh Lewsey retained at full-back following his superb two-try display against Italy.
With Cohen fit again following a thigh strain, Dan Luger and James Simpson-Daniel - Woodward's wings in the Italy game - drop out.
"I am very pleased with the side," said Woodward. "I am delighted to welcome back Martin Johnson, Neil Back, Jason Leonard, Jason Robinson and Ben Cohen to the starting line-up.
"They have all made excellent progress from their injuries and it is a compliment to the England and club medical teams working in partnership, that they have made a full recovery so quickly."
He added: "We are going game by game, and this is not the time to experiment too much. The 6 Nations is a very important tournament.
"I don't take Scotland lightly - they've created a lot of positive rugby this season."
Leonard, who suffered a hamstring injury while winning his 100th cap against France last month, returns instead of Robbie Morris at tighthead prop, and Back reclaims the openside flanker's role, so Richard Hill reverts to blindside and there is no place for Joe Worsley.
There are several changes on the bench, including a place for Northampton fly-half Paul Grayson, who won the last of his 23 caps during World Cup 99, while Andy Gomarsall is preferred to Kyran Bracken as scrum-half cover.
England team to play Scotland at Twickenham:
Backs: J Lewsey (Wasps), J Robinson (Sale Sharks), W Greenwood (Harlequins), M Tindall (Bath), B Cohen (Northampton), J Wilkinson (Newcastle), M Dawson (Northampton);
Forwards: G Rowntree (Leicester), S Thompson (Northampton), J Leonard (Harlequins), M Johnson (Leicester, capt), B Kay (Leicester), R Hill (Saracens), N Back (Leicester), L Dallaglio (Wasps).
Replacements: M Regan (Leeds), T Woodman (Gloucester), D Grewcock (Bath), J Worsley (Wasps), A Gomarsall (Gloucester), P Grayson (Northampton), D Luger (Harlequins).