Leicester took the lead after 16 minutes, with Ricardo Pereira surging forward before shooting superbly into the bottom-left corner.
Newcastle were reduced to 10 men just before half-time when Isaac Hayden was sent off for a dangerous challenge on Dennis Praet.
Jamie Vardy made it 2-0 on 54 minutes, beating Martin Dubravka at his near post with a powerful low shot.
Leicester’s third came three minutes later, with Paul Dummett diverting Praet's centre into his own net.
Vardy scored his second on 64 minutes, heading in substitute Marc Albrighton’s left-wing cross, before Wilfred Ndidi fired in the fifth in the 90th minute.
A fourth win in five matches puts Leicester in third place on 14 points.
Newcastle remain 19th with five points.
Leicester: 5 Oct v Liverpool (A)
Newcastle: 6 Oct v Man Utd (H)