Halep has twice finished runner-up in Rome in 2017 and 2018, and "really wants to win" the tournament.
"It's one of the biggest goals now," Halep, who is on a run of 12 consecutive victories after winning in Dubai in February and in Prague after the lockdown, said.
She will play Spaniard Garbine Muguruza, the No.0 seed, for a place in the semi-finals after the Spaniard ousted US Open finalist Victoria Arazenka.
World No.2 Halep dominated before No.30 Putintseva withdrew after 45 minutes citing a lower back injury, clearing the path to Muguruza.
"It's a big challenge," Halep said of meeting the 2016 French Open winner, who like the Romanian has also won Wimbledon.
"Muguruza, she feels really good on clay court and here, as well."
Muguruza leads Halep 5-2 in previous meetings but both the Romanian's wins were on clay.
No.9 seed Muguruza has reached the semi-finals in Rome three times, but the Spaniard has never advanced to the finals.
Halep has twice finished runner-up in Rome in 2017 and 2018 to Svitolina.
Halep skipped the US hard court tour because of coronavirus concerns, but has extended her winning streak this season to 12 consecutive victories.
She won the Dubai title in February, before the coronavirus lockdown, coming back to claim victory in Prague last month.
Also into the last four is No.2 seed Karolina Pliskova, who beat No.11 seed Elise Mertens, from Belgium, 6-3 3-6 6-0.
"I don't know which year it can be. For sure there is a chance," Pliskova, a former French Open semi-finalist, said of winning Roland Garros this year.
"I think definitely it was maybe one of the best matches this season, but it's not so difficult because there was not many matches.
"I felt great. Especially the first set I think was maybe one of my best sets what I have played on clay ever."
Pliskova next plays Marketa Vondrousova, last year's French Open runner-up, having beaten her fellow Czech in their previous clash at the Miami Masters last year.
"She's a strong opponent who can play quite an ugly game on clay," former world No.1 Pliskova said.
"It's semi-finals, still kind of preparation for Paris. If I play like today I think I should not be worried."