Jeremy Hunt took a swipe at the right-winger as the battle to seize control of the Tories boiled over.
Shadow chancellor Mr. Hunt warned that elections are “always won from the center ground” after the former Home Secretary launched an astonishing attack on party moderates.
Ms Braverman, who is expected to run to be Tory leader, used a weekend interview to warn that the Conservatives risk becoming “centrist cranks”.
But appearing on the BBC's Sunday With Laura Kuenssberg programme, Mr Hunt, who has ruled out running himself, said that the next Tory leader had to restore the Tory reputation for “calm competence”.
“In a two-party system, in the UK as we have, elections are always won from the center ground,” he said.
But I think the really interesting question is what is the center ground in British politics?
“And I think ordinary, decent British families want controls on migration, they want the government to show restraint on spending and keep taxes down, they want welfare to be reformed so it's fair to people who get up early and work hard.
“I think that is the center ground, that is where the Conservatives should be and we need to carve out a position that does that, but reflects honestly on the fact that we didn't get everything right and we need to be open about why. we didn't get everything right.
“And that is the way we will give people confidence that we'll be able to do things differently than before.”
Lord Houchen, the elected Tory mayor of the Tees Valley, had a similar message, telling Sky News that the party needed a “back to basics leader” and urged Ms Braverman to conduct this leadership contest with “civility”.
Ms Braverman has previously suggested that the Conservatives should welcome Nigel Farage into the party.
She used an interview with the Telegraph at the weekend to suggest that the Tories are refusing to learn the lesson of the general election.
“If we don't recover the voters we deliberately, and arrogantly, spurned, we will turn the Conservative Party into the 21st century version of the 20th century Liberal Party,” she said.
“And we can do better than being a collection of fanatical, irrelevant, centrist cranks, who make it our business to insult our should-be voters for not being as smug and self-righteous as we are.”
Ms Braverman has denied reports she is considering defecting to Reform UK, after repeatedly attacking her Tory colleagues.
The Conservative Party is expected this week to set out a timetable and process for choosing Rishi Sunak's successor, after an election which left it with just 121 MPs.
Former security minister Tom Tugendhat has received the endorsement of two prominent former Tory MPs in a boost to his hopes.
Damian Green and Steve Baker, who lost their seats in the July 4 election but are influential figures in the party.
Mr Baker is an ardent Brexiteer, and former leader of the European Research Group, while Mr Green was chairman of the One Nation Group of Tory moderates.