She held off a brief challenge from Tsurenko in the second set before reeling off the last five games of the match to improve her head-to-head record against her opponent to 6-0.
World number three Halep, who received a bye in the opening round, will next face ninth seed Julia Goerges.
The German battled past American lucky loser Alison Riske 6-1 6-7(5) 6-4 in a marathon encounter, recovering from 4-2 down in the final set.
Fourth seed Elina Svitolina began her campaign with a 6-4 6-4 win over 2017 French Open champion Jelena Ostapenko, avenging her two previous losses to the Latvian.
A neck injury has hampered the Ukrainian, who was playing her first match since the Australian Open quarter-final defeat by eventual champion Naomi Osaka of Japan in January.
"I have been struggling since the Australian Open with my neck," Svitolina said after the match.
"It depends... on the practice, on the match, what I have, so it's tough to tell how I'm going to feel tomorrow, but every single day I try to get a treatment, I try to get some work done, and, you know, it's just a process."
There were moments in the match when Ostapenko was unplayable as she unleashed aggressive forehands from the baseline to recover from early breaks of serve in both sets.
But Svitolina maintained her service rhythm to ensure she controlled the match in the critical moments and went on to convert five break points en route a convincing victory.
"The ball is flying and the wind is very tricky, so I had to be extremely quick with my feet. In the end I was very pleased that I could finish in two sets," added Svitolina, who will meet Czech qualifier Karolina Muchova in the quarter-finals.
Earlier on Wednesday, Czech Karolina Pliskova was forced to withdraw from the tournament due to a viral illness.
The 2017 Doha champion is the fifth withdrawal since the draw was made following Danish sixth seed Caroline Wozniacki (viral illness) and France's Caroline Garcia (leg injury).
(Reporting by Hardik Vyas in Bengaluru; Editing by Ken Ferris)
