Iheanacho scored a brace as the Foxes overcame Fleetwood Town in round three and did likewise at London Road after debutant Fousseni Diabate opened the scoring.
Former Manchester City striker Iheanacho netted his only senior hat-trick against Aston Villa in this round two years ago and his nine goals overall is more than any other player has managed since he made his January 2016 bow in the competition.
Diabate, a reported £2million signing from Ligue 2 club Gazelec Ajaccio earlier this month, also shone and cut a lovely finish across Peterborough goalkeeper Jonathan Bond in the ninth minute.
Iheanacho arrowed a fine left-footed strike into the bottom-left corner despite Harvey Barnes screaming for a pass in space outside him, effectively ending the contest three minutes later.
The Nigeria international then doubled his tally by steering in a volley from Christian Fuchs' sensational left-wing cross after 29 minutes.
Andrew Hughes grabbed a reward for Peterborough's spirited second-half response from a 58th-minute corner, but the waves of Leicester attacks kept coming and Diabate crowned an excellent showing by firing into the roof of the net three minutes from time after Bond saved from Demarai Gray.
Wilfred Ndidi finished in similar fashion two minutes into stoppage time to complete the rout.
Key Opta Stats:
- Leicester have progressed from seven of their last eight FA Cup fourth-round ties against lower-league opposition, with their only defeat coming at the hands of West Brom in 2001-02.
- Peterborough have failed to keep a clean sheet in any of their 32 FA Cup matches against top-flight opposition.
- Andrew Hughes' strike was the first competitive goal that Leicester have conceded in 508 minutes of action in 2018.
- Kelechi Iheanacho has been directly involved in 10 goals in nine FA Cup appearances (nine goals, one assist).
- Since his FA Cup debut in January 2016, Kelechi Iheanacho has scored more goals in the competition than any other player in this period (9).