Melbourne's Fourlis, the world No.147, was up against a player ranked more than one hundred places higher in southern Italy on Tuesday and lost in 75 minutes.
Putintseva will face Diane Parry in the round of 16 after the Frenchwoman defeated American Lauren Davis 7-6 (7-5) 6-2.
World No.39 Putintseva is now the leading player in the draw after top-seeded Martina Trevisan withdrew with a back injury,
Italian Trevisan was replaced by lucky loser Carolina Alves of Brazil, who will face Elina Avanesyan later in a repeat of their match in the final round of qualifying.
Fifth-seeded Caroline Garcia beat fellow Frenchwoman Chloe Paquet 6-7 (2-7) 6-2 6-2 and will meet Italian wildcard Elisabetta Cocciaretto in the next round after she eliminated childhood hero Sara Errani 6-1 6-0 in less than an hour.
At the Hamburg Open, top seed Anett Kontaveit beat Irina Bara 6-3 7-6 (7-3) despite losing her serve four times, and next plays Sweden's Rebecca Peterson.
Estonian world No.2 Kontaveit started the season strongly but has struggled since a COVID-19 infection in April and was 1-3 in her previous three tournaments, including the French Open and Wimbledon.
Third-seeded Czech Barbora Krejcikova, the 2021 French Open champion, beat Poland's Magdalena Frech 6-3 6-0 in the second round and will face either Anastasia Potapova or Maria Lourdes Carle in the quarter-finals.
German eighth seed Andrea Petkovic beat Misaki Doi of Japan 6-4 6-3 and will play either Kontaveit or Peterson in the last eight.
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