Moyles made the announcement that he would be leaving the programme live on air this morning.
27-year-old Grimshaw is currently Radio 1's late-evening DJ from Monday to Thursday. In addition, he co-hosts the Sunday evening 10pm-to-midnight show with Annie Mac.
He is also a TV presenter on Channel 4's morning weekend show T4.
Grimshaw said today: "I'm super excited to be hosting the iconic Radio 1 Breakfast Show, it's been a dream of mine since the age of 11 and to be honest it hasn't really sunk in yet. I love Chris and have always looked up to him as one of the best broadcasters ever."
Ben Cooper, Controller BBC Radio 1 and BBC Radio 1Xtra, said: "Nick is a great broadcaster with a passion for music and a mischievous sense of humour, which has made him a hit with our listeners. He will be an excellent host for Radio 1's new breakfast show, bringing a new generation of listeners with him."
Grimshaw joined the BBC in 2007, presenting shows on BBC Two and Radio 1, for the BBC's former teen strand. He then went on to present the station's weekend breakfast show from September 2008 before moving to his current show. Originally from Oldham, Nick's first step into broadcasting was on student radio at Liverpool University.
Other presenters touted for the role included Greg James, Fearne Cotton and Reggie Yates.