The Lichfield MP has urged his parliamentary colleagues to stick with Rishi Sunak, despite the Tories losing more than 1,000 councillors as Labour, the Lib Dems and Greens made gains.
He said national issues were largely to blame for the result in Lichfield, where the Conservative group lost its majority by a single seat after losing seven councillors.
But he said the PM still has time to turn things around – providing he addresses key issues including the cost of living crisis.
Mr Fabricant told the Star: "The main thing Rishi needs to do is that he has to deliver more money in people's pockets. If people feel they are worse off than they were before, then we won't get elected.
"But if we try and get rid of another Prime Minister, we will not only not win the next election, we won't even deserve to win the next election. We get no Brownie points with the electorate for appearing to be like rats fighting in a sack."
He said the Lichfield result – which saw the Conservatives finish with 23 seats, ahead of Labour on 17 and the Lib Dems with three – was "hugely disappointing".
But he said with the party remaining the biggest on the council, he hopes they will "be able to continue to deliver for the people of Lichfield".
"Many good Conservative candidates have lost mainly as a consequence of the turmoil in Westminster last year and particularly because of the current cost of living crisis," Mr Fabricant added.
"This has been echoed in elections right across Europe. My heartfelt commiserations go out to hardworking councillors who have lost their seats not because of anything they failed to do, but because of a national trend. And I congratulate all the councillors who have been elected of every political party.
“As the MP for Lichfield, I will continue to work closely with the council to help residents in Lichfield, Burntwood, and the rest of the constituency."
At Friday's count at Burntwood Leisure Centre, the Tories needed two seat for a majority as the Whittington and Streethay ward went to recounts.
Richard Holland won won of them, but Staffordshire County Council leader Alan White, who had held the seat for 20 years, lost out to the Lib Dem candidate by three votes.