The motion, submitted by the centre-right opposition, failed to garner the required 233 votes in parliament where Grindeanu's left-wing Social Democrat party (PSD) holds a solid majority after winning elections only two months ago.
The prime minister had appeared defiant ahead of the vote and vowed not to quit.
"We have a duty to the people who gave us their trust during the election and to continue to govern," the 43-year-old told lawmakers in Bucharest.
Although the measure was scrapped late Sunday, the marches have continued, with some protesters vowing not to stop until the government steps down.
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While the crowds have noticeably shrunk from the half a million people thronging cities and towns on Sunday -- the largest rallies since the fall of communism in 1989 -- they are expected to grow again over the weekend.
"Every action the government took in the last week proves that they are not honest at all. So we cannot trust them," protester Danchiric, who works in advertising, told AFP at Bucharest's Victory Square where 3,000 people had gathered on Tuesday night.
Observers say much of the public anger is directed at the graft-riddled political establishment, which includes powerful PSD head Liviu Dragnea.
The 54-year-old was barred from running for office because of a voter fraud conviction and is currently on trial for alleged abuse of power, a charge he denies.
"The government has understood the demonstrators' message. Other measures will be taken to end this conflict," Dragnea said on Wednesday.
In a parliamentary address Tuesday, Iohannis had hinted that the government should quit.
"The repeal of the decree and the possible sacking of a minister is too little. Early elections are too much," Iohannis said.
"If the PSD, which has created this crisis, fails to resolve the crisis immediately, I will summon all the political parties for talks. You've won, now govern and legislate -- but not at any price," he warned.
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