United got off to the perfect start, breaking the deadlock in the seventh minute with a bullet of a strike from Kim Jae-sung - against his former club.
However, the lead lasted just a minute with Magno Cruz equalising for the home side with a towering header.
Not to be denied, the Reds restored their lead when Dylan McGowan nodded home from a corner on 49 minutes before teenage midfielder Riley McGree rounded out the scoring on 65 minutes.
The win moved Adelaide above Jeju in Group H courtesy of a better head-to-head record. Both sides have four points. Group leader Jiangsu Suning, nine points, and Gamba Osaka, three points, are in action in the group's other matchday four encounter.
The goals flowed early in Jeju with Sergio Cirio capitalising on a defensive error. When his shot cannoned back off the upright the rebound found its way to Korean Kim, and his first time striker from 20 metres found the net for his first goal in a Reds shirt.
The A-league side's joy was short-lived when Lee Chang-Min collected the ball at the byline and looped a cross to the back post where Magno Cruz outjumped Ben Garuccio to nod his header past Eugene Galekovic.
Jeju dominated the rest of the half and carved out chance after chance but a combination of sloppy passing and desperate Reds defending kept the score level.
The final few minutes of the first half saw both sides enjoy clear chances to take the lead with Mendy directing his header just past the post before Adelaide almost scored.
Cirio's corner was met by McGowan whose header forced a strong save from Jeju keeper Lee Chang-Geun. Cirio reacted quickest and immediately cut the ball back into the penalty area for Taylor Regan but he too was denied by Chang-Geun who again was equal to the task, keeping out the close range strike.
Jeju burst out of the interval and attacked immediately from the restart but it would be the Reds who would find the net four minutes in to the second half.
McGowan outjumped two defenders to powerfully head home a deep corner from Garuccio, with the goalkeeper in no man's land.
Minutes later Jeju wasted another chance when substitute Jin Seong-Uk blasted wide from 10 metres with the Reds defence scrambling to thwart the attack.
Adelaide extended their advantage in impressive fashion with a well-crafted third goal, started and finished by McGree.
He linked with Baba Diawara on the edge of the area, who then fed Kim who cut the ball back for McGree whose initial attempt was saved. McGree then scooped the ball into the net while laying on the deck.
Galekovic kept the score at 3-1 with a stunning reflex save late on, denying Seong-Uk from two metres.
The Reds custodian was called in to action numerous times in the final minutes as Jeju peppered Adelaide's goal looking to salvage a result.