The Croatia striker has fallen out of favour at the Merseyside club and West Ham have been linked with a move for him after boss Roberto Martinez revealed he will not stand in the former Rangers man's way if an acceptable offer is received this month.
Jelavic made the most of his first start since September as he scored a goal in each half at Goodison Park and he also missed a penalty as Everton brushed aside their Championship opponents.
The impressive Ross Barkley opened the scoring and Seamus Coleman took his tally for the season to six goals with another stunning strike as Everton ran riot.
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 line-up, 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-foot finish 10 minutes before the break.
Everton doubled their lead just before half-time, when Karl Henry was caught in possession and Jelavic fired home the loose ball with a fine left-foot strike from 25 yards out.
QPR gave the ball away again in midfield and Coleman should have punished them, but the overworked Cesar made a fine fingertip save 11 minutes after the break.
Everton looked dangerous every time they attacked and Jelavic capitalised 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 were awarded a penalty after 74 minutes when Oviedo was brought down by Cesar.
Jelavic went for the cheeky chip for his hat-trick, but it back-fired as he struck the crossbar.
However, Everton were not to be denied a fourth goal two minutes later when Coleman beat Cesar with another a sublime right-foot finish as Rangers crumbled.
Steven Naismith and Jelavic both had efforts cleared off the line by Clint Hill as Rangers were totally outclassed.