The motion for an inquiry, proposed by Greens Senator Rachel Siewert, was passed with the help of Labor and independent Senator Nick Xenophon.
Centrelink sent tens of thousands of letters to people before Christmas 2016, claiming that their earnings were at variance with what they had declared to the Australian Taxation Office.
They were given up to three weeks to explain, or told they would face an adverse assessment and have to pay a recovery fee of 10%.
GetUp national director Paul Oosting called for the debt recovery programme to be scrapped altogether.
"An inquiry will answer the question on everyone’s minds – how did a government-driven system get it so wrong?" he said.
"But in the coming months, as the Senate investigates, false debt letters will continue to be printed by the minute, and the Turnbull Government will still be attacking tens of thousands of Australians with threatening false debt letters."
Oosting offered the case study of a nurse named Jenny, aged 54, who lives in Revesby Heights, a suburb of NSW, and has two grandchildren.
"She severed the tendons in her thumb in an accident at home, and went to the hospital where the laceration was glued," he said.
"She was unable to work as a result of the injuries, a situation compounded by being given the wrong treatment, and required five additional surgeries to rectify the mistakes. During part of this time, she received sickness benefits from Centrelink for approximately nine months.
"Worried she was going to lose her house, she rang Centrelink to find out whether she could perform casual work in a computer role. Hours worked and monies earned were reported to Centrelink each time."
Oosting said eventually a settlement was paid by the hospital's insurance company "due to negligence with the original treatment she received".
"Despite being out of work for 15 months, she received no further benefits from Centrelink, but in November, she received a letter for $4121.44 debt. She called Centrelink to explain that the money she had received as sick benefit had been repaid to Centrelink by the insurance company as part of winning her negligence case, but there was no record of this on hand and not considered.
"Hours into her second call with Centrelink, she was informed that she was able to appeal the decision, but the debt management team give her no explanation for her false debt.
"In January, she received letters from Centrelink saying she has no debt, and letters dated the same day from the debt collectors saying her payments are overdue."
Oosting said Prime Minister Malcolm Turnbull had the power to pull the plug on the programme.
"Based on the number of brave Centrelink whistleblowers and wrongly targeted people that have appeared in the media so far, we're predicting this Senate inquiry is going to be very uncomfortable for Malcolm Turnbull and his government," he said.
"But instead of seeing sense and shutting down the robo-debt scam, Malcolm Turnbull is continuing to let his government bully pensioners and people living with disabilities for cash they don't owe."