A 3-1 loss at West Brom earlier on Saturday saw the Gunners remain in fifth place and the thumping of the Tigers has closed the gap to the Toffees to mere goal difference.
Lukaku, who added two goals late on to punish Hull, on Tuesday questioned whether Everton had the ability to reach the Champions League.
His double on Saturday, with both goals in added time, may not have been the decisive factor in putting the Toffees level on points with fifth-placed Arsenal, but it at least showed his professionalism cannot be called into question.
The platform had been laid by a first goal for 20-year-old Dominic Calvert-Lewin, in his sixth Premier League appearance, and, after Tom Huddlestone's red card, another from Enner Valencia. Lukaku then took centre stage to register his 20th and 21st league goals of the season, the first time an Everton player has reached that tally since Gary Lineker in 1985-1986.
The win puts Everton tantalisingly close to the top four, five points behind Liverpool with the Merseyside derby at Anfield up next after the international break.
Lukaku's comments angered Koeman, but the Dutchman knew only too well dropping the Premier League's top-scorer would amount to cutting off his nose to spite his face.
However, for nearly 80 minutes an unconvincing performance against relegation strugglers appeared to be adding weight to the striker's argument.
Lukaku always had Curtis Davies or Harry Maguire - two of Hull's three centre-backs - in close attention and, apart from scuffing his first attempt and being flagged offside as he stroked a shot past Eldin Jakupovic, he was barely involved in the first half.
The Belgian picked up after the break, with one shot rifled over from the edge of the penalty area and another fierce angled drive tipped behind by Jakupovic, which had the Gwladys Street End singing his name.
He did, however, finally produce a significant contribution with a lobbed assist for Valencia to score Everton's 78th-minute second within a minute of coming off the bench before confidently beating Jakupovic twice in added time.
The manner in which one grateful fan gleefully accepted Lukaku's shirt when he hurled it into the crowd at the final whistle showed his harsh words have not affected his popularity - helped of course by his goals.
With all the focus on the club-record signing's future, it was a trio from the club's youth teams who claimed the spotlight with the eighth-minute opener.
Barkley's piercing through-ball inside wing-back Andrew Robertson picked out Tom Davies, timing his run perfectly, and his low cross was side-footed home by Calvert-Lewin, who has emerged from the under-23s this season after arriving from Sheffield United in 2016. It was his first start since early January after injury.
But the pace dropped when Gareth Barry replaced the injured Morgan Schneiderlin and Everton were almost caught napping when Ashley Williams' hesitation in dealing with a long ball saw Sam Clucas' lob land on the top of the net just before half-time.
Koeman switched formation to match Hull's three at the back as the visitors began to gain the upper hand, only for Huddlestone's dismissal for a shin-high challenge on Idrissa Gana Gueye to halt their momentum in the 73rd-minute.
It released the pressure building up on the hosts, who quickly added the second to finally kill off the game before Lukaku stole the spotlight once again.