Halep, the defending champion from Romania, had been forced to go the distance in the first two rounds.
Playing her best match of the tournament, she had no trouble against an error-strewn Tsurenko, winning 6-2, 6-1 in just 55 minutes to extend her win streak against the No 27 seed from Ukraine to 8-0.
After a relatively cool start to the first week of the French Open, temperatures are expected to soar to as high as 30 degrees Celsius (86 F) at the end of the afternoon at Roland Garros in Paris.
Taking advantage of the balls kicking up higher on the clay of the main Court Philippe-Chatrier, Halep took the first set in just 28 minutes as she dominated her opponent from the back of the court.
Tsurenko, who had won her previous match 11-9 in the decider against Serbia's Aleksandra Krunic playing with a broken racket, requested a medical timeout as she trailed 3-0 in the second set.
Watched by Romanian billionaire Ion Tiriac, Halep's intensity never dropped, as she took a 4-0 lead with aggressive shot-making. Serving for the match at 5-0, Halep got broken, but booked a spot in the fourth round in the next game with a volley, her ninth winner.
She made 14 errors, while Tsurenko had just 8 winners and made 32 unforced errors.
"It was a tough match even if the scoreline does not show it," Halep said. "Every day in my life I have that image of me winning last year in my mind, it was my greatest moment of my career and it was all the more satisfying for having won here in the juniors as well."
Zverev through to fourth round
It took Alexander Zverev all five sets again as he advanced into the fourth round of the French Open with a 6-4, 6-2, 4-6, 1-6, 6-2 victory over Serbian Dusan Lajovic on Saturday.
The German fifth seed, who fought his way through a five-set thriller in the opening round, threw away a two-set lead against the 30th seed on Court Simonne Mathieu before setting up a meeting with Fabio Fognini.
It was a highly inconsistent performance by the lanky 22-year-old Zverev, who will need to sharpen his focus against the Italian ninth seed.
Fognini progressed with a 7-6(5), 6-4, 4-6, 6-1 victory against Spanish 18th seed Roberto Bautista Agut.
Zverev did not make things easy for himself as he fell 3-1 behind in the opening set before managing to turn things around and take the lead.
He raced through the second only to find himself on the back foot again in the third and fourth as Lajovic forced a decider with temperatures flirting with 30 degrees Celsius — a contrast with the chilly weather in the first week of the tournament.
After an early trade of breaks in the fifth, Zverev finally got his confidence back and finished it off with a whipping forehand, his 52nd winner barely making up for a woeful 55 unforced errors.
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