The 20-year-old academy product was outstanding as high-flying Leicester overcame their hosts 4-1 in their second-round tie at Gigg Lane.
Ranieri made 11 changes and gave Dodoo, a forward or winger, his chance after being impressed by him in a youth game.
Ranieri said: “I am very happy with him. He deserved to show his quality and tonight made a very good job of it.
“I watched him once with the young players and he scored three goals. I wanted to play with wingers. I put him on the right and it was a good choice. It was a fantastic performance.”
Leicester eased through the first half with Dodoo opening the scoring and Andrej Kramaric adding a second. Bury rallied after the break and replied through Danny Mayor but their hopes were killed off as Dodoo grabbed two more in the closing four minutes.
Ranieri said: “The second half was a battle and we showed very good spirit. It was important to be involved in a battle and go through in the cup.”
Prior to the game, there had been some suggestion Dodoo might be made available for loan.
Ranieri said it was now “important to reflect and wait” before making a decision but opposite number David Flitcroft joked that he would be welcome to join the Shakers.
Flitcroft said: “We have identified some potential targets and I have got a new one now in the boy who played against us on the right-hand side.
“I am just going to fill Claudio with some red wine and get him signed up tonight!”
Flitcroft was disappointed his side could not force an equaliser as they held the upper hand for periods in the second half.
He said: “For people who have not been in the stadium and see 4-1, it looks like a completely overpowering performance from Leicester. But it is just the way things conspired. We were really in the game at 2-1.”
Flitcroft claimed Dodoo’s second goal in the 86th minute was “offside by two yards” and killed the contest.
That goal also sparked ugly scenes as a number of Bury fans ran towards the side of the pitch, apparently in the direction of the assistant referee. One Leicester player was dragged away as stewards moved in.