The House of Commons leader, speaking on LBC about the public inquiry report into the fire, had said ignoring the fire service's stay-put policy "just seems the common sense thing to do".
Jeremy Corbyn called for Mr Rees-Mogg to apologise and Grenfell United, a group that represents survivors and bereaved families of victims, criticised his comments as "beyond disrespectful".
In a statement responding to criticism, the Commons Leader said: "I profoundly apologise.
"What I meant to say is that I would have also listened to the fire brigade's advice to stay and wait at the time. However, with what we know now and with hindsight I wouldn't and I don't think anyone else would.
"I would hate to upset the people of Grenfell if I was unclear in my comments."
Boris Johnson's spokesman said following Mr Rees-Mogg's "full apology" that the prime minister has "full confidence" in his Commons leader.
Conservative MP Andrew Bridgen has been criticised for defending Jacob Rees-Mogg over his controversial comments about the Grenfell Tower victims.
Mr Bridgen told BBC Radio 4's PM show that the Commons leader would have made a "better decision" than the authority figures who were giving the victims advice on the night of the tragedy.
Andrew Gwynne, Labour's national campaign co-ordinator, said Mr Bridgen's comments were "contemptible" and that he should be removed as a parliamentary candidate.
And grime rapper Stormzy said Mr Rees-Mogg should resign over the comments.
On Twitter the music star said "these politicians are actual aliens" for the way Grenfell residents have been spoken about.
In a flurry of tweets Stormzy called Mr Rees-Mogg a "scumbag".
While leaving 10 Downing Street Mr Rees-Mogg was asked by ITV News Political Correspondent Paul Brand if he believed the residents of Grenfell "lack common sense".
The Commons leader responded: "That's not what I said."
Following his apology Conservative colleagues came to his defense, with MP Nigel Evans claiming Mr Rees-Mogg "is an absolute asset to the Conservative Party".
He added: "I think that as leader of the House he's done a tremendous job."
Tory MP Bill Cash denied that Mr Rees-Mogg's comments made their party seem "nasty".
He said: "I'm afraid that that's one way of some people putting it but it's not the case and it's certainly isn't the case that we are in anyway a nasty party."
His apology comes after sharp criticism from Mr Corbyn and shadow housing minister, Sarah Jones.
The Commons leader had said on Monday: “If you just ignore what you’re told and leave you are so much safer.
“I think if either of us were in a fire, whatever the fire brigade said, we would leave the burning building.
“It just seems the common sense thing to do, and it is such a tragedy that that didn’t happen.”
The Labour leader said Mr Rees-Mogg should apologise for the “crass and insensitive comments immediately”.
Mr Corbyn said: “What possesses someone to react to an entirely avoidable tragedy like Grenfell by saying the victims lacked common sense?
“People were terrified, many died trying to escape.”
His sentiments were echoed by Ms Jones.
She said: “These are appalling comments.
“Jacob Rees-Mogg should retract them and apologise immediately.
“Statements like this just go to show how out of touch the Tories are and are a glaring admission of their failure to act in the interest of the Grenfell victims and their families.”