"Lawrence has been a fundamental part of some of the most successful British and English rugby," said McGeechan.
"He is a natural leader. He was very important back in 1997, he has been fundamental to England's success over the last eight years and his club side has been going well with his leadership. He has been central to some very successful rugby.
"He has set examples and the best thing we can do now is use examples he has set to take things forward. That is the sort of thing he would want."
That process starts immediately as McGeechan announced that Martin Corry will captain the Lions against Taranaki in New Plymouth.
"You can't replace a character like Lawrence Dallaglio. He's been immense. Just his presence around the squad is brilliant to have," said Corry.
"What has happened is incredibly unfortunate and as a whole squad we have got to take this on the chin and move on.
"It is not about looking over your shoulder and saying what might have been. We can all make excuses but soft men make excuses and we have got to get our heads down and get on with it."
New Zealand's assistant coach Steve Hansen offered his sympathies and admitted that, despite being in the opposition camp, he was hugely disappointed that Dallaglio will not feature in the Test series.
"I think it was really sad. It has robbed the tour of a personality, and it has robbed the tour of a great player," he said.
"I know he is in the opposition but I really feel for the bloke. He made the sacrifice of not playing for England to make it on this tour and he gets injured in the first game, that is just bad news.
"If you lose a world class player it weakens you and it must be devastating for him.
"It is also sad for the New Zealand public because they are going to miss out on seeing a great player at his best."