The 21-year-old Nigerian opened the scoring with only his second goal for the club in the 43rd minute as Islam Slimani played a neat ball over the top and Iheanacho coolly converted.
He was similarly sharp with 13 minutes remaining as he latched onto Riyad Mahrez’s through ball and lifted it precisely over Fleetwood goalkeeper Chris Neal.
But his celebrations were to be temporarily halted by the linesman’s flag.
The goal was awarded, however, 68 seconds after it hit the net as video official Mike Jones told Moss that Pond’s back foot had kept Iheanacho onside.
The second goal effectively ended Fleetwood’s resistance and set up a fourth round match between Leicester and League One side Peterborough United.
West Ham had been held to a goalless draw at Shrewsbury in the first tie and needed 112 minutes of the replay to break the deadlock at the London Stadium.
The goal was worth waiting for, however, as Burke met a cross and thrashed the ball into the top corner off the underside of the bar leaving West Ham to play Bournemouth or Wigan Athletic in the next round.
Cardiff swatted League Two Mansfield aside 4-1 away to set up a tie at home to Premier League leaders Manchester City.
The English Premier League champions’ manager Pep Guardiola was watching the third round replay from the stands.
Sheffield Wednesday scored in each half through Marco Matias and Atdhe Nuhiu to beat visiting League Two outfit Carlisle 2-0.
Reading’s Icelandic forward Jon Dadi Bodvarsson got a hat-trick for the second-tier side who beat Stevenage 3-0.
