Paul scored 14 of his 28 points in the fourth quarter as Western Conference top seeds Phoenix broke open a close game, withstanding a 35-point performance from Mavericks star Luka Doncic.
There was less drama in Miami, where the East's top-seeded Heat again capitalized on the injury absence of Philadelphia star Joel Embiid as they rolled to a 119-103 victory over the 76ers to take a 2-0 lead in their best-of-seven series.
Dallas, trying to head home for games three and four with a split of two games in Phoenix, took a 60-58 lead into halftime.
But the Mavs had no answer for Paul, who keyed a 23-2 fourth quarter scoring run that effectively sealed it for Phoenix with minutes to spare.
Devin Booker scored 21 of his 30 points in the second half for the Suns, who connected on a franchise playoff record 64.5% of their shots from the field. That included 52% from three-point range as the Suns drained 13 baskets from beyond the arc.
"(We) just had to be a little bit more aggressive," Paul said. "That second quarter we started turning it over. They got it going. Shooters were making shots and we just wanted to stay the course."
Booker helped launch the turnaround with back-to-back three pointers early in the third quarter, but the Suns' 89-83 lead heading into the final period of the physical, sometimes testy contest looked tenuous.
Paul, however, took control, making six straight shots to gut-punch Dallas.
"We just did what we were supposed to do," Paul said as he looked to games three and four in Dallas starting on Friday. "We won two games at home."
That was the same sentiment voiced by Heat star Jimmy Butler in Miami.
"We did what we were supposed to do at home," Butler said. "The series really starts when you take it to the road."
Bam Adebayo scored 23 points and pulled down nine rebounds and Butler added 22 and 12 assists for Miami.
Tyrese Maxey scored a game-high 34 points for Philadelphia, Tobias Harris added 21 and James Harden had 20.
But three-time scoring champion Harden was held to just four points in the second half, as the Heat, who trailed briefly in the first quarter but led 60-52 at halftime, comfortably kept the Sixers at bay.
- Not enough shots -
Tyler Herro scored 16 of his 18 points in the first half off the bench and made three of his five three-point attempts for Miami.
Victor Oladipo came off the bench to score 19 points and was an even more efficient three-of-four from beyond the arc as the Heat made 14 three-pointers.
The Sixers, meanwhile, struggled from three-point range, connecting on just eight of their 30 attempts.
"When it comes down to it, you've got to make shots, especially on the road," Harden said. "We got enough stops when we need to, had numerous chances. You know (if) we make a couple of shots that gives us more confidence and it puts more pressure on them.
"I feel like we never really put that pressure on them throughout the course of the game."
The series resumes on Friday, and it's not clear when Embiid will return after suffering a concussion and orbital fracture in the series-clinching win over Toronto.
"He's got so many steps to go through," Sixers coach Doc Rivers said. "I don't think he's cleared any of them right now. So we just have to wait and see."