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Mexican outfielder Edgar Gonzalez allowed the runners to score after he bobbled Rizzo's flyball in front of a crowd of 4,478 at Salt River Fields stadium in Scottsdale, Arizona.
Rizzo hit his double off Sergio Romo, who is the closing pitcher for the defending World Series champion San Francisco Giants.
Jorge Cantu hit a bases-loaded double to right field to give Mexico a 3-2 lead in the first inning after Alex Liddi's two-run single had put Italy ahead.
Mexico's Ramiro Pena added a run-scoring double in the second to boost Mexico's edge to 4-2, but Drew Butera blasted a two-run homer for Italy in the fourth to equalize at 4-4.
Luis Cruz put Mexico back on top 5-4 in the fifth with a line drive to centrefield scoring Eduardo Arredondo from second base.
Italy is trying to get out of the first round of the WBC for the first time after being eliminated early in 2009 and 2006.
Group C opening-round play was set to begin later Thursday with Venezuela facing the Dominican Republic at Puerto Rico.
Two teams from each group advance to the second round at Miami next week.
Meanwhile, Canadian infielder Brett Lawrie pulled out of the event on Thursday because of a strained left rib cage.
Lawrie got hurt while fielding a ball at third base during Team Canada's exhibition game against the Cincinnati Reds on Wednesday.
Canada starts their pool play against Italy.