Mr Trump tweeted to thank Gen McMaster, saying he had done an “outstanding job & will always remain my friend”.
Mr Bolton, who has backed attacking North Korea and Iran, told Fox News his job would be to ensure the president has “the full range of options”.
Gen McMaster is the latest high-profile departure from the White House.
Last week, Mr Trump fired Secretary of State Rex Tillerson by a tweet, replacing him with former CIA Director Mike Pompeo.
Mr Bolton, whose appointment does not require US Senate confirmation, will be Mr Trump’s third national security adviser in 14 months. He will take the job on 9 April.
Mr Bolton, 69, has been a foreign policy hawk in Republican circles for decades, having served in the administrations of Ronald Reagan, George HW Bush and George W Bush.
The second Bush appointed him as US envoy to the UN, during which time diplomats privately criticised Mr Bolton’s style as abrasive.
A strident neo-conservative, Mr Bolton helped build the case that Saddam Hussein possessed weapons of mass destruction, which turned out to be wrong.
Known for his walrus moustache, Mr Bolton does not appear to have greatly moderated his views since his last spell in government.
He stands by the invasion of Iraq and has advocated in newspaper op-eds using military force against North Korea and Iran.
In a brief statement on Thursday, Gen McMaster thanked President Trump for appointing him and said he was applying to retire from the US Army later this year.
The 55-year-old three-star general is leaving after just over a year as national security adviser.
The White House said Mr Trump and Gen McMaster had “mutually agreed” that he would leave. He had been rumoured for weeks to be on his way out. (Courtesy BBC)