The 66-year-old, who lost her Glasgow seat in the Commons at the 2015 election, had previously served as shadow Scottish secretary.
She had previously been an MSP at Holyrood and held a number of posts within the Scottish executive.
Curran wore the traditional scarlet robes for the short introduction ceremony in the unelected chamber, where she swore the oath of allegiance to the King.
She was supported by fellow Labour peers Helen Liddell and Harriet Harman.
Also taking his seat in the unelected House was former MEP for London Claude Moraes.
The 59-year-old previously served as deputy leader of the European Parliamentary Labour Party and chaired the European Parliament’s Civil Liberties, Justice and Home Affairs Committee.
Dressed in the scarlet robes, Moraes was supported by leader of the Lords Angela Smith and chief whip in the upper chamber Roy Kennedy as he swore the oath of allegiance to the monarch.
Prime Minister Keir Starmer was elected to the Labour leadership on a pledge to abolish the House of Lords, but has since U-turned and has instead opted for minor reform, removing hereditary peers from the chamber.