Valentina Giacinti and Aurora Galli netted either side of the interval to set up a meeting with the Netherlands or Japan, who play later on Tuesday (1900 GMT).
Italy have made it to the last eight of the women's World Cup only once, in the inaugural edition in 1991 when the group phase directly led to the quarter-finals.
China, the 1999 runners-up, enjoyed a good spell before the break but failed to make it count as Italy keeper Laura Giuliani made some good saves.
"It's an incredible result in a difficult game. Today was not a great match, but football is not just that... the girls gave everything," said coach Milena Bertolini.
"Reaching the quarter-finals is a huge satisfaction. Now we need the warmth of the Italians, because the more we go on, the more the level increases."
"We’ve done a good job. We need to do much better. We couldn’t win the game and I apologise to all the Chinese fans," said China coach Jia Xiuquan.
"We understood Italy very well and it was a good game. This is football. Whoever makes mistakes first, has to pay.”
Bertolini's team were focused from the outset and were rewarded after 15 minutes as Giacinti started and finished the move for Italy's opener.
She broke down the right flank and her cross was met by Barbara Bonansea, who played in Elisa Bartoli.
The fullback's attempt was parried by China keeper Peng Shimeng into the path of Giacinti, who had followed up to stab home from close range.
The Azzurre went close to doubling their tally when Valentina Bergamaschi's fierce angled shot forced Peng to save at full stretch.
China had a couple of chances through Wang Yan but Italy keeper Giuliani stayed alert to preserve her team's lead, as the pressure on her goal increased thanks to some fine creative play by Wang Shuang.
Italy, however, regained the momentum early in the second half when Galli, who had replaced the injured Cristiana Girelli in the first half, found the back of the net with a low 20-metre strike in the 50th minute.
It was her third goal of the tournament, with all three having been scored after she came off the bench.
China pushed to find a breakthrough but Italy's back four were largely untroubled as they kept their opponents at bay until the final whistle.
(Reporting by Julien Pretot; Editing by Toby Davis and Pritha Sarkar)
