Morton takes over from Edward Argar, who had led on youth justice policy and remains as a junior minister in the Ministry of Justice.
As parliamentary under secretary of state for the Ministry of Justice her brief includes overseeing family law and justice, tackling domestic abuse, miscarriages of justice as well as youth justice matters.
She takes up the role after serving as assistant government whip from January 2018 to the end of July this year.
Morton tweeted she was "honoured" to take the role.
She has been an MP, for Aldridge-Brownhills, in the West Midlands, since 2015 and first joined government in 2016, as a junior business, energy and industrial strategy minister.
She moved to a similar ministerial role at the Department for International Development a year later.
As an MP, Morton has consistently voted against lowering the voting age from 18 to 16, a total of three times in 2015.
Argar's role sees him focus on areas including supporting victims, sentencing and human rights.
He will also support the government in preparing for Brexit. Both Argar and Morton were opposed to the UK's exit from the European Union in the 2016 referendum on the issue.
Argar joined the MoJ in 2018 as a junior minister and like Morton, became an MP 2015.
Before moving roles within the MoJ Argar, who is MP for Charnwood, Leicestershire, ruled out ending youth custody.
During a parliamentary debate in June he said that depriving a child of their liberty should be used "as a last resort" but backed the focus on education at future secure schools for young offenders, the first of which is planned for Medway.