The Foxes boss changed 10 players, with just Gokhan Inler remaining from the team that started the 2-2 Premier League draw at Stoke on Saturday, but so fluent were his side that you would not have known it.
Ranieri's men were on the front foot for much of the third-round tie, creating numerous chances, but found Hammers goalkeeper Adrian on top form.
Academy product Joe Dodoo marked his full home debut with the opening goal after six minutes, his fourth in the competition after his hat-trick at Bury in the previous round, but West Ham equalised before the half-hour mark through Mauro Zarate's deflected shot.
Leicester spurned numerous opportunities to win in the second half but time after time they were denied by Adrian.
The Spanish goalkeeper continued to frustrate the Foxes during extra-time before Andy King finally broke his resistance with a terrific header with 116 minutes on the clock.
"We made a fantastic performance
I think we made 10 great chances to score, their keeper was man of the match," said Ranieri
"I am very pleased, we were fantastic
We changed 10 players but they played at the same level
They're a great group and they work very hard
I'm very happy
"Some of the players have not been used in the Premier League but I believe in them
I told them that I want to go through the rounds and I pick you to play because I believe you can win
"Everybody was fantastic tonight
King scored the goal but they were all fantastic.
"It is important to remain unbeaten this season to stay confident everybody and for everybody to show their quality."
Ranieri admitted he thought the game was destined for a penalty shoot-out.
He said: " We had he names of the first five penalty takers written down
But we deserved to score a winner
It was a great goal."
West Ham boss Slavan Bilic admitted Adrian had kept his team in the game but the Croatian felt the manner of the defeat was cruel
"He made some good saves, especially in extra-time," said Bilic
"I think he did brilliantly in the two one-on-one situations
He kept us in the game in extra time
"They started better than us but then after that for the normal 90 minutes we were a bit better than them
But in extra time we started to get tired and when you concede a goal with four minutes left you are very disappointed
"We are very disappointed because we approached this competition with the aim to win
"We knew it would be tough, though
We had a tough draw against another Premier League side who are playing really well and we are away from home, it was always going to be hard
"To concede so late is tough, disappointing."
Source : PA
Source: PA