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.
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.
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 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.
