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|Ons Jabeur serves to Simona Halep in their Dubai clash. AFP/KARIM SAHIB|
"I don't know how I was able to finish this," said Halep who had to save a match point herself before progressing.
"It was up-and-down because she doesn't give rhythm. It was very weird, but I'm happy I could win."
The Wimbledon champion's win came after Karolina Pliskova and Garbine Muguruza re-established order among the top seeds after a run of upsets at the Aviation Club.
World number three Pliskova strolled to a 6-1, 6-2 defeat of qualifier Kristina Mladenovic in barely an hour, before ninth seed Muguruza took almost two-and-a-half hours to get past Russian Veronika Kudermetova 7-5, 4-6, 6-4.
Three big names had exited the day before -- defending champion Belinda Bencic, third seed Elina Svitolina and Australian Open winner Sofia Kenin.
The tense Halep match resembled a Fed Cup tie, with rival groups of Tunisian and Romanian fans out-shouting each other.
"It was like a football match or a Fed Cup match. I didn't really realise what was going on on court," added Halep.
"Everyone was with a lot of energy so we had to give our best. I think it was a great match."
Wildcard Jabeur, ranked 45, came from 4-2 down in the final-set tiebreaker, missing a match point of her own when she netted a forehand.
Halep, who lost the opening set in 26 minutes, eventually claimed victory when Jabeur drove long.
"It was a very tough match, she played well," Halep said. "I wanted to give all I had and just wanted to win. I didn't give up, I'm proud I fought to the end."
In the quarter-finals, Halep faces Belarusian seventh seed Aryna Sabalenka who defeated doubles partner Elise Mertens of Belgium 6-4, 6-3.
Pliskova pegged back Mladenovic after the Frenchwoman had earned only her first main-draw win of the season in the first round.
"It's always a tough one against her. We played in Australia, so I knew what to expect," Pliskova said.
"I was not playing amazing but still, even with a couple of mistakes, I didn't give up. The score looked a little bit easier than it really was, we had a couple of tough games."
Muguruza went into her match two days after beating Kim Clijsters in the Belgian's first WTA match since 2012.
"There's a lot of things that I could have done better," two-time Grand Slam champion Muguruza said.
"But I'm excited that I gave myself another opportunity to be in the quarters."
Last month's Australian Open finalist added: "I'm just happy that I get these tough matches.
"Honestly, the difference is very, very small, I'm just pleased to go through."
Muguruza will bid for a semi-final place on Thursday, taking on Jennifer Brady, a 4-6, 6-4, 6-1 winner against 2019 Roland Garros runner-up Marketa Vondrousova.
Simona Halep (ROU) bt Ons Jabeur (TUN) 1-6, 6-2, 7-6 (9/7)
Aryna Sabalenka (BLR) bt Elise Mertens (BEL) 6-4, 6-3
Jennifer Brady (USA) bt Marketa Vondrousova (CZE) 4-6, 6-4, 6-1
Garbine Muguruza (ESP) bt Veronika Kudermetova (RUS) 7-5, 4-6, 6-4
Petra Martic (CRO) bt Barbora Strycova (CZE) 6-3, 6-3
Anett Kontaveit (EST) bt Anastasia Pavlyuchenkova (RUS) 7-6 (7/5), 7-5
Elena Rybakina (KAZ) bt Katerina Siniakova (CZE) 6-3, 6-3
Karolína Pliskova (CZE) bt Kristina Mladenovic (FRA) 6-1, 6-2