Raggi, a 37-year-old lawyer whose campaign tapped into simmering discontent over a string of corruption scandals in the Italian capital, won between 33 and 38 percent of the vote, according to exit polls by public broadcaster RAI and La7 television.
She was trailed by Roberto Giachetti of Renzi's centre-left Democratic Party (PD), who secured 20 to 26 percent of the vote.
A second round run-off is to take place on June 19.
Eyes were also trained on the mayoral election in Milan, where centre-left candidate Giuseppe Sala took the lead with 38 to 42 percent, in line with pre-poll expectations, according to La7.
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He was closely followed by the centre-right's Stefano Parisi.
Overall, more than 13 million people nationwide were asked to cast their ballots to choose members of 1,300 municipal councils in a two-round ballot.
The Italian capital has been without an elected leader since last October, when Ignazio Marino, a member of PD, was forced to quit over an expenses scandal.
Even if Raggi doesn't win, her strong showing is likely to boost the populist Five Star movement, led by comedian Beppe Grillo, as it seeks to cement its status as a mainstream party.