Dallaglio has been discharged from hospital after undergoing surgery to have a plate and five screws inserted into his right ankle.
The former England captain dislocated it, broke his fibia and damaged ligaments when he slid into a tackle and was caught awkwardly under a pile of bodies in the first half of the Lions' 34-20 victory over the Bay of Plenty on Saturday.
Sir Clive Woodward immediately sent for Easterby, who was already at Shannon airport about to fly out to Japan with Ireland when the call came in from the Lions.
"I am not coming here to fill Lawrence's boots, he is a one-off," said Easterby on his arrival in Auckland.
"Looking at the game on Saturday he was in outstanding form before he got injured, he looked really sharp.
"It is a huge blow for him and a real disappointment, but I will hopefully bring something of my own to the tour and make a positive impact when I get the chance."
Dallaglio will return home in about a fortnight, once the stitches have been removed from his leg, but he believes the Lions can go on to be victorious in New Zealand in his absence.
Dallaglio's experience and rare leadership qualities - not to mention the prime form he has been in - made him a central figure to the Lions' hopes of glory in New Zealand.
However, Dallaglio predicted Easterby could yet have an important role to play.
"I am very disappointed that I cannot take any further part in the tour because I felt that I had a lot to offer," he said.
"I wanted to be part of achieving something very special over here and I believe the resolve of the team will be strengthened by what has happened.
"There are a number of players who will now have to step up in terms of leadership but we do have that kind of quality in the squad.
"Players will rise to the challenge and that's the beauty of this kind of tour, players relish the chance to put themselves forward.
"Players will come to the fore in, at times, the most peculiar of situations.
"Whenever you lose anyone from the squad it's going to be a setback but it's a fact that one person's misfortune is another person's opportunity.
"Simon Easterby has come out here after a fantastic Six Nations and has a really good opportunity."
Easterby was not aware he was top of the Lions' standby list.
"I was not expecting it but I was hopeful that if there was an unfortunate case where someone got injured and was out of the tour, that I would be in the frame to come out here," he added.
"The squad has been together for a few weeks now and I have only come in today. There is a lot I need to full my mind with."