Hull claimed their first ever win over Liverpool as the visitors missed the chance to close on leaders Arsenal.
The hosts went ahead when Jake Livermore's shot took a wicked deflection off defender Martin Skrtel.
A precise free-kick from Steven Gerrard pulled Liverpool level but David Meyler capitalised on some poor defending to put the hosts back in front.
Luis Suarez fired wide but the Reds rarely threatened before Skrtel's headed own-goal ended their hopes.
Allam, who took over in 2010, infuriated some supporters with his idea to rebrand the club Hull Tigers.
The Hull fans have formed a campaign group called City Till We Die.
But Allam told the Independent: "They can die as soon as they want, as long as they leave the club for the majority who just want to watch good football."
Rather than ease the tension with supporters the remarks have been seen as extreme in many quarters and there was no let up at Sunday's game.
The 74-year-old Egypt-born businessman, who moved to Hull in 1968, first announced in August the company name had changed from Hull City AFC to Hull City Tigers, ending a part of the Yorkshire club's name which has been in effect since their formation in 1904.
Allam later stated a Tigers brand would be more marketable, and considered the word "City" to be "lousy" and "common".
A number of supporters protested during the game against Crystal Palace last week by unfurling a banner which read "We are Hull City".
But the owner added: "How can they call themselves fans, these hooligans, this militant minority, when they disturb and distract the players while taking away the rights of others to watch the football, and of companies who have paid good money for advertising?"
City Till We Die hit back at Allam's remarks.
"Thankfully, hooliganism at football in this country is now exceedingly rare," the group said in a statement.
"The intemperate suggestion that singing "City Till I Die" or holding a banner with Hull City's name on it constitutes disorder is ill-informed, unhelpful and will be considered by many to be offensive.
"We reiterate our advice to all City fans to continue their fine support for our fantastic team while positively expressing a preference for our current name."
When Hull were relegated from the Premier League in 2010 they were thought to be approximately £35m in debt but Allam insists he will walk away from the club immediately if the criticism of his proposals continues.
The owner has already offered to refund season ticket holders who are unhappy with the name changes.