The Reds scored two goals in a six-minute span in the first half, with Argentine Jeronimo's long-range bullet followed by a near tap-in for Spaniard Sergio Cirio.
Adelaide's Portuguese winger Fabio Ferreira scored in the second stanza with a sweet strike before he fed Jeronimo his second goal.
Wellington's Kenny Cunningham pegged one back before Adelaide's Kenyan-born winger Awer Mabil netted in injury time.
Adelaide climbed to third on the ladder, though Melbourne Victory can re-take that spot with a win or draw in Saturday night's derby against Melbourne Heart.
The Reds' form is compelling: losers just twice in their past dozen matches, they took command of Wellington from the outset at Coopers Stadium.
Jeronimo's blast to open scoring was from the top shelf, the attacker cutting inside a Phoenix defender and, from a bouncing ball, cracking a thunderous 15-metre right-footer - the pure power gave Wellington keeper Glenn Moss no chance.
Six minutes later, Cirio was the beneficiary of slick build-up work from Ferreria.
After slicing through the defence, Ferreria's cut-back pass found the Spaniard unmarked five metres from goal and he duly converted.
Adelaide's dominance continued in a second half marred by serious injury to Wellington's Reece Caira, who suffered a suspected broken cheekbone when inadvertently struck by a Tarek Elrich elbow as they contested a header.
After Caira wobbled off, Adelaide rapidly added two goals - a swivelling Ferreria cleanly struck a first-timer from 10 metres to score in the 59th minute.
And three minutes later, Ferreria delivered a perfect cross to Jeronimo who slotted his second from close-range.
The Phoenix struck back in the 67th minute when Cunningham beat the keeper with a 15m shot but it was too little, too late for the visitors, who remain in seventh spot.
Adelaide United coach Josep Gombau said the win was most satisfying.
"We are playing well, I am so happy," he said.
"Just keep going. I don't want the players to relax."
Wellington Phoenix's Caira has a fractured eye socket and their star attacker Cunningham may have broken a foot late in the match.
"When things go wrong, they go wrong properly," said Phoenix coach Ernie Merrick.