Mr Trump recruited David Perdue, an ex-senator, to challenge incumbent Brian Kemp in the Republican primaries.
Mr Kemp had rejected Mr Trump’s pleas to overturn the 2020 Georgia election results.
But as ballots were counted, Mr Perdue was so far behind that he was defeated before half the votes were in.
The primary was being closely watched as a test of Mr Trump’s hold over the Republican party, as voters decide who will be the party’s official candidate in the midterm elections in November.
The midterms will decide who controls the two chambers which make up Congress – the Senate and the House of Representatives – and fall halfway through President Joe Biden’s term in office.
The result in Georgia sets up a general election rematch between Mr Kemp and Democratic candidate Stacey Abrams, who had accused the Republican of suppressing votes in her previous tilt at the governor’s seat in 2018.
In a speech after the results, Mr Kemp said: “Our battle is far from over. Tonight the fight for the soul of our state begins to make sure that Stacey Abrams is not going to be our governor”.
Allegations of unfair voting practices are likely to emerge again in the crucial state, where a highly competitive Senate race is also due to take place in November.
Democrats wrested the state’s two Senate seats from Republicans in 2021 – including one from Mr Perdue – to gain a slim majority in the upper chamber of the US Congress.
However, one senator, Raphael Warnock, must now defend his post against Herschel Walker, a well-known former American football player, who is also backed by Mr Trump.
But the governor’s race was seen as the bigger test of the former president’s clout.
Mr Kemp, a staunch conservative, drew the former president’s ire after he certified that his state had been won by Joe Biden in 2020. Mr Trump had alleged that there were 11,000 votes for him to be found which would reverse the result, but the governor disagreed.
Georgia’s top election official, Secretary of State Brad Raffensperger, who also refused Mr Trump’s entreaties at the time, won his bid for re-election on Tuesday.
Mr Trump’s claims that the election had been “stolen” from him has become a divisive issue within the party, splitting “establishment” Republicans from Trump populists, even dividing the former president’s own administration members.
In the primary, Mr Kemp was backed by Mr Trump’s former deputy, Vice President Mike Pence.
Separately, Mr Trump’s preferred candidate for Senate in Pennsylvania is also facing the prospect of defeat.
The Republican primary race between Mehmet Oz, a celebrity doctor backed by the ex-president, and David McCormick, a former hedge fund CEO is so tight that ballots are to be recounted, state officials said.
Tallies show the two candidates separated by just 902 votes, or approximately 0.1%.
Under state law, a recount must take place if the candidates are separated by 0.5% or less.