Maurizio Lupi, who had been resisting calls to quit since the scandal first broke earlier this week, confirmed to Italian television that he would step down on Friday after answering questions about the case in parliament.
Lupi maintained that he had done nothing wrong, saying he was stepping down purely to avoid further damage to the government's credibility.
"I believe my gesture will strengthen the action of the government," he said in a pre-recorded interview with Porta a Porta, the flagship news and discussion programme of public broadcaster Rai.
"I have not lost my honour, nor my passion," he added, saying he hoped to be able to continue his political career.
Lupi, 55, denied that he had been pressed to quit by Prime Minister Matteo Renzi, as has been widely reported.
The minister, and by extension Renzi, had been embarrassed by the steady dripfeed of damaging reports following the announcement on Monday of an investigation into the suspected rigging of tenders for major public works projects worth a total of €25 billion ($26 billion).
Among four men arrested in a probe in which a total of 51 officials and businessmen were placed under investigation, was Ettore Incalza, a senior official in Lupi's department until the end of last year.
Lupi had been particularly embarrassed by the revelation that a businessman who was one of Incalza's alleged accomplices had given his son, Luca Lupi, a €10,000 Rolex watch.
Examining magistrates suspect that the same businessman, Stefano Perotti, organized a job for the recently-graduated son at an engineering firm run by his brother-in-law, at the request of the minister.
Lupi has described Perotti as a family friend and wiretap evidence leaked to the press suggests he was also close to another businessman, Francesco Cavallo, who has also been arrested in the kickbacks case.
Several recordings feature Cavallo introducing himself to business contacts as "Lupi's man."
Others appear to indicate that he arranged for hand-made suits to be made for the minister and some of his closest aides, as well as buying flights for Lupi's wife.
When the job issue first emerged, Lupi responded by insisting that he would never seek favours for his children and that to do so "would have been a grave error and, I presume, a crime."
Twenty-four hours later a recording emerged of a phone call in which he is heard telling Incalza: "You must come and meet my son."
Judges suspect Incalza asked Perotti to arrange things and, within a month, Luca Lupi was a site manager on a Milan building site which already had 17 managers.
Renzi, who has made a more effective fight against corruption in Italian life a central plank of his reforming premiership, was reported to have demanded that Lupi resign immediately.
The centre-left leader was unable to simply sack the minister as it would have risked destabilising his coalition government. Lupi is a leading member of the New Centre Right (NCD), the junior partner in a coalition dominated by Renzi's Democratic Party (PD).
The investigative judges in charge of the kickbacks inquiry believe the four arrested men were at the helm of a system which enabled them to skim off one to three percent of the value of contracts for high-speed rail links, new metro lines in Rome and Milan and other huge projects.
The scandal is the biggest of its kind since the "tangentopoli" cases of the early 1990s.
That episode led to the indictment of half the country's lawmakers but, two decades later, Italy is still struggling to shake off a reputation for corruption that economists say has cost the country billions in lost investment.