The top seed, who won the Cincinnati final a year ago over Jessica Pegula, increased her lead at the top of the WTA Tour match-win statistics as she secured her 49th of the season.
But the struggle was real for Sabalenka, who finished with two aces in the closing stages.
"I'm happy to get through this difficult match," Sabalenka said. "I just hope tomorrow is a day off. I need time to recover from such a great battle."
Raducanu put up a huge fight, matching Sabalenka throughout.
The British number one survived a 25-minute eighth game of the final set, which went to 13 deuces, with Raducanu saving four break points before finally holding for 4-4 and eventually battling into a tiebreaker.
But Sabalenka squeezed out victory with a sixth ace for 5-4 in the breaker and delivered her seventh on second match point to bring the battle to a close after three hours and nine minutes.
"It was all about momentum," Sabalenka said. "I took some risks at the end, went for crazy shots, went to the net.
"A risky game from me helped to put a lot of pressure on her," she said of an opponent whom she hugged warmly at the net.
Sabalenka is bidding for her 10th trophy at the 1000 level and second this season after winning at Madrid.
Elsewhere, former champion Madison Keys booked her place in the last 16 with a 6-4, 6-0 win over Japan's Aoi Ito, edging a tight first set before storming through the second in just 20 minutes.
Next up for the Australian Open champion is Kazakh ninth seed Elena Rybakina, who recovered from a first-set wobble to battle to a 4-6, 6-3, 7-5 win over Belgian Elise Mertens.
"In the first set I struggled a lot. I think I never did so many double faults. Something for me to learn. Happy in the end I managed to find my game. Thank you so much guys for staying so late," the 2022 Wimbledon champion said after beating Mertens.
Anna Kalinskaya stunned American fifth seed and Wimbledon runner-up Amanda Anisimova 7-5, 6-4 for her third win against a top-10 opponent in the season. Kalinskaya will face fellow Russian Ekaterina Alexandrova in the next round.
In other third-round results at the last major tuneup prior to the US Open, Ekaterina Alexandrova defeated Australian Maya Joint 6-4, 6-3.
Sorana Cirstea defeated Yuan Yue 6-7(2), 6-4, 6-4 and will next play third seed Iga Swiatek, who got a walkover into the round of 16.
Spain's Jessica Bouzas Maneiro, a quarter-finalist this month in Montreal, made further hard-court progress with a 6-4, 6-1 thrashing of American wildcard Taylor Townsend.
Her next opponent will be Sabalenka on Wednesday.