Councillor Steve Tulley made national headlines last week after he claimed Ms Cooper, "wouldn't know democracy if it scratched her in the eyeballs" in an explosive attack on her attitude to Brexit.
The Pontefract, Normanton and Castleford MP was criticised by Leave supporters for helping to take a 'no deal' Brexit off the table with a Parliamentary amendment last month, though her actions drew praise in other quarters.
Coun Tulley made his remarks after being re-elected to serve South Elmsall and South Kirkby on Wakefield Council, on a night when three of his Labour colleagues lost their seats.
He said that Ms Cooper had caused the local authority "problems" and that Westminster should "take notice" of the 2016 referendum result, in which two thirds of voters in the district voted to quit the EU.
But Coun David Jones, the chair of the Normanton, Pontefract and Castleford Constituency Labour Party, said that Ms Cooper was protecting jobs in the area and that she had the complete backing of her local branch.
Coun Jones, who held on to his own Pontefract South seat in last Thursday's polls, said: “We were incredibly sad to lose hardworking colleagues (at the elections) and I want to, on behalf of the constituency Labour Party (CLP) thank them for all the hard work they have done over the years.
"Our CLP supports Yvette’s work that she had been doing, to try to get a workable Brexit deal for jobs and prevent a chaotic no deal instead.
"We have 15,000 jobs in manufacturing and distribution in this constituency. Yvette is standing up for every single one of them and we are behind her.
"She was elected to stand up for our area and that’s exactly what she’s done - avoiding a no deal nightmare that would have pushed up food prices for hard working families and would have hit the worst off hardest."
Ms Cooper, who has not responded to the comments herself, attended the election count at Thornes Park Stadium on Thursday night, but left the arena before Coun Tulley made his remarks.
Couns Glenn Burton (Knottingley), Yvonne Crewe (Airedale and Ferry Fryston) and Glyn Lloyd (Hemsworth) all lost their seats to the Liberal Democrats and two independent candidates, respectively.
The local authority's leader Peter Box admitted it had been a "disappointing" night for the ruling Labour group as their number of elected members was cut from 52 to 49.