Tottenham looked all set to put their name into the hat for the fifth round when Andros Townsend opened the scoring from the penalty spot in the 19th minute after Liam Moore fouled Roberto Soldado.
However, Leicester produced a late rally as first Leonardo Ulloa levelled in the 83rd minute, before Jeffrey Schlupp side-footed home an injury-time winner.
Manager Pearson urged Leicester to take the momentum forward and move off the foot of the table.
"Winning is important in any competition, especially in the situation we're in," he said.
"We had to dig in, but we made some luck. We kept going and we needed to. Tottenham are a squad full of athletes.
"They are also very good technically and as a unit physically they are pretty impressive, so for us it was a game in which we had to compete on lots of different levels.
"In the first half when they had lots of possession we had to show a lot of courage to stick in there."
Pearson was not entirely happy, though, as he believes Andrej Kramaric should have been awarded a penalty by referee Robert Madley just before the break when he went to ground under a challenge from Spurs goalkeeper Michel Vorm.
The striker instead receiving a caution for simulation and Pearson believes technology should be use to clarify controversial decisions.
"Those sorts of issues do need addressing," he added. "It is very easy for managers to forget about it, based on the result.
"But we certainly felt aggrieved that we didn't get a penalty - and on top of that Kramaric being booked was harsh.
"It's another example of why don't we use technology? It wouldn't take long to clarify the situation."
