"The hateful language... from public figures on TV does not represent the true spirit of this country," she said, to loud cheers and applause.
"Don't stoop to their level. Our motto is, when they go low, we go high."
Earlier, Vermont Senator Bernie Sanders urged Democrats to back Mrs Clinton.
She will accept the party's presidential nomination on Thursday, when a roll call of how states' delegates are voting is to be read out.
"While Donald Trump is busy insulting one group after another, Hillary Clinton understands that our diversity is one of our greatest strengths," Mr Sanders said.
Some supporters of the senator, who had been Mrs Clinton's rival in the Democratic primary, booed at mention of Mrs Clinton's name; others wore duct tape with the word "silenced" over their mouths.
Mrs Obama's speech was widely seen as an attack on Mr Trump, although she did not mention the Republican nominee by name.
She said she wanted her husband to be succeeded by "somebody who knows this job and takes it seriously" and that could only be Mrs Clinton.
Watch: Michelle Obama speech in full
Because of the Democrats' candidate, she said, "my daughters and all our sons and daughters now take for granted that a woman can be president of the United States."
In other remarks, she praised Mrs Clinton as a "true public servant" who had shown "devotion to our nation's children" and "never buckles under pressure".
After her speech, President Obama wrote on Twitter: "Incredible speech by an incredible woman. Couldn't be more proud & our country has been blessed to have her as FLOTUS [first lady]. I love you, Michelle."
Mr Trump, who has criticised his rivals, both Republican and Democrat, in sometimes colourful language on Twitter, said Mr Sanders had "sold out" to Mrs Clinton.
He also condemned other speakers at the Democratic convention including senators Cory Booker and Elizabeth Warren, but made no mention of Mrs Obama.
In other key moments from the convention on Monday:
The Washington Post said the party's "opening day of discord" ended on a "high note" but USA Today said the Democrats proved "they can be dysfunctional too".
The Boston Globe meanwhile asked: "Has Bernie Sanders lost control of his delegates?"
But it was the First Lady who was credited with invigorating things. This was Hillary's convention, but it was "Michelle's night" says Gail Collins in the New York Times.
The LA Times says she "delivered one of the most passionate Clinton endorsements of the night".
1. What is the point? At a convention, each party formally nominates its candidates for president and vice-president, and the party unveils its party platform, or manifesto.
2. Who is there? There are about 5,000 delegates attending, selected at state and congressional district conventions, and representing each US state and territory. Plus about 15,000 journalists and tens of thousands of other party grandees, lawmakers and guests.
3. What is the schedule?
US first lady Michelle Obama has denounced Republican Donald Trump and passionately backed Hillary Clinton at the Democratic National Convention in Philadelphia.