Cipriani lit up Twickenham on Saturday through a dazzling display on his full Test debut as England brushed aside RBS 6 Nations opponents Ireland 33-10.
Wilkinson was dropped following England's Calcutta Cup defeat against Scotland a week earlier, and now clearly faces a major battle to reclaim the number 10 shirt.
Dallaglio, Cipriani's club captain at Wasps, hailed the 20-year-old's performance.
But he also has no doubt Wilkinson will respond in the right fashion.
Dallaglio told BBC Radio's Five Live Sportsweek: "I think it was a fantastic start for Danny.
"Given the pressure of the situation that he came under - he was the only change for Jonny Wilkinson in the whole England team - so it was kind of 'this was the guy who was going to make the difference.'
"He played incredibly well. He did the simple things right and he did them very effectively.
"He looked like a man who had been playing international rugby for a long time, which is great for Danny and great for England that we've got a competitive situation in that position now.
"He was a guy making his first full start for England. I can remember my first start for England, and I was as nervous as anything. The way Danny controlled and ran that team was flawless, really.
"I don't think Jonny will be nervous about it.
"We all get to a stage in our careers when the challenge becomes a different one to the one it was before, and Jonny has been a very, very special player and still continues to be a very special talent.
"The challenge is taken either way. You either lie down and let someone else take your place, or you rise to the challenge, and I have no doubt Jonny will go away and work on the areas of his game that he feels he needs to improve.
"Jonny earnt his reputation as an unbelievable player who tackled incredibly well, goal-kicked immaculately and had the ability to control the game.
"I think where Danny differentiates himself is he is prepared to play flatter and really get the back-line moving, which certainly showed yesterday."