The Croatian striker scored his first two goals of the season as Roberto Martinez's side blew away Harry Redknapp's men to secure a spot in the fourth round of the FA Cup
England youngster Ross Barkley struck an impressive opener from 15 yards out after being set up well by Bryan Oviedo on 35 minutes, as the Premier League side dominated Harry Redknapp's Championship hopefuls.
Jelavic deservedly doubled the Toffees' lead a minute before the break with his first goal of the season by firing past Julio Cesar from 20 yards, and grabbed his second of the game with a close-range finish on 68 minutes.
The Croatian missed the chance to score a hat trick by hitting the crossbar with a penalty after being felled by Nedum Onuoha with 16 minutes remaining, but Seamus Coleman took his tally for the season to six goals when his thunderous effort from the right-hand side almost broke the back of the net 60 seconds later.
Martinez won the FA Cup with Wigan Athletic last season and will be hoping for a repeat performance in a competition he described as "unique" on Friday.
The Spaniard made five changes to his starting lineup, with Jelavic coming in for his 50th start for the club. Goalkeeper Julio Cesar was handed his first QPR start since the two sides last met at Goodison Park in April.
Rangers fashioned the first chance of the game when midfielder Gary O'Neil tried his luck with a volley from long range and was narrowly off target.
Everton enjoyed plenty of possession, but O'Neil should have put the visitors in front when he failed to connect with Danny Simpson's inviting cross.
The home side raised the tempo and Leon Osman spurned a great chance when he fired over the crossbar after Cesar had palmed away Barkley's effort from a tight angle.
Barkley was brimming with confidence and the midfielder showed why he is so highly regarded as he cut in from the left and found the bottom corner of Cesar's net with a clinical right-footed finish 10 minutes before the break.
Everton doubled the lead just before halftime, when Karl Henry was caught in possession and Jelavic fired home the loose ball with a fine left-footed strike from 25 yards out.
Everton looked dangerous every time it attacked and Jelavic capitalized on some poor defending to convert Bryan Oviedo's cross from close range after 68 minutes.
The outstanding Barkley then burst forward on a stunning solo run but fired over the crossbar, and Everton was awarded a penalty after 74 minutes when Oviedo was brought down by Cesar.
Jelavic went for the cheeky chip for his hattrick, but it backfired as he struck the crossbar.
However, Everton was not to be denied a fourth goal two minutes later when Coleman beat Cesar with a sublime right-footed finish as Rangers crumbled.