Commentators noticed similarities with Mrs Obama’s 2008 convention speech.
Mrs Trump’s speech also praised her husband as a “compassionate” man who would “fight for the country”.
But Mr Trump’s team said Mrs Trump used “common words” and blamed Hillary Clinton for the row.
“There’s no cribbing of Michelle Obama’s speech,” said Paul Manafort, Mr Trump’s campaign manager.
“I mean, this is, once again, an example of when a woman threatens Hillary Clinton, how she seeks out to demean her and take her down,” he added.
It was Mrs Trump’s first speech of the campaign and she had the help of a speechwriting team.
Republican National Committee Chairman Reince Priebus he did not blame Mrs Trump, but added he would “probably” fire the speechwriter over the controversy.
Earlier, Mr Trump’s communications adviser Jason Miller issued a statement saying a “team of writers” used her life’s “inspirations” as source material.
Meanwhile Mr Trump himself described her speech as “absolutely incredible”.
The Slovenian-born former model had used the speech to portray her husband as a tough but compassionate unifying force.
“He’s tough when he has to be but he’s also kind and fair and caring,” Mrs Trump said.
“The kindness is not always noted, but it’s there for all to see. That is one reason I fell in love with him to begin with.”
Mrs Trump has kept a lower profile compared with other spouses of major candidates. The Trumps’ adult children have often been more prominent.
Mrs Trump’s mostly personal speech contrasted with the night’s other speakers who spoke about “making America safe again”.
Each day of the convention is based on a similar theme. Tuesday’s focus will be “make America work again”, with House Speaker Paul Ryan, UFC President Dana White and New Jersey Governor Chris Christie taking centre stage.
A wide range of speakers, including a county sheriff and a decorated combat veteran, told stories of military service, urban crime and the dangers of illegal immigration on Monday.
Former New York City Mayor Rudy Giuliani and Arkansas Senator Tom Cotton took aim at presumptive Democratic nominee State Hillary Clinton, questioning her judgement in foreign affairs.
“Who would trust Hillary Clinton to protect them?” said Mr Giuliani, whose rousing speech fired up the crowd at the Quicken Loans Arena in Cleveland, Ohio. “I wouldn’t.”
Many Republican party leaders – such as Ohio Governor John Kasich whose state is hosting the convention – have decided not to attend this year’s festivities, turned off by Mr Trump’s controversial stances on immigration and foreign policy.
Mr Trump has called for a ban on Muslims entering the US and also advocated deporting almost 11 million undocumented immigrants.
A speech by Wisconsin police sheriff David Clarke strongly defended police officers in the wake of the deadly shooting in Baton Rouge, which killed three officers over the weekend.
“Blue lives matter!” Mr Clarke said to huge cheers from the delegates in the convention hall.
The killings prompted Mr Trump to say the country was falling apart – a claim strongly disputed by President Barack Obama.
Earlier, anti-Trump delegates failed in their final push to block Mr Trump’s nomination.
The vote that would have allowed delegates to back a candidate of their choice was quashed when three states reportedly backed out.
Some said the Republican Party officials had sabotaged their efforts on purpose.
The push was the last gasp for those who hope to see a different Republican candidate get the nomination.