Murray served at 4-5 in the third set, fell behind 15-40 in that game, saved both match points, then faced another break point in the same game and again survived.
Sonego led 3-0 in the decisive tiebreaker before Murray won the next six points to pull away and win the match.
With the tournament's top four seeds receiving first-round byes in the 28-player draw, seventh-seeded Alejandro Davidovich Fokina (23) was the top seed in action on Monday. The Spaniard beat France's Constant Lestienne (30) 6-4, 6-3.
Other first-round winners were Great Britain's Liam Broady and Australians Jason Kubler and Christopher O'Connell.
Open 13 Provence
Switzerland's Stan Wawrinka (37) registered the only two service breaks of the match while beating Belgium's Zizou Bergs (23) 6-4, 6-7 (6), 6-4 in the first round at Marseille, France.
Wawrinka got the only break in the opening set in the final game and the only break of the third set in the first game. The three-time Grand Slam champion saved all seven break points on his own serve.
In the opening day's only other two main-draw matches, Kazakhstan's Alexander Bublik (25) came from behind to beat Serbia's Filip Krajinovic (30) 5-7, 6-3, 6-4, and Benjamin Bonzi (26) topped Luca Van Assche (18) 7-5, 7-5 in an all-French clash.
Rio Open
Brazil's Thiago Monteiro (28) saved the only four break points of the third set in the opening games, then went on to defeat Austria's Dominic Thiem (29) 6-1, 3-6, 7-6 (2) in the first round at Rio de Janeiro.
Monteiro improved his ATP Tour singles record to 2-6 this year, while Thiem, the 2020 US Open champion, fell to 1-5.
Ninth-seeded Alex Molcan (25) of Slovakia easily handled Brazilian wild-card entrant Joao Fonseca 6-0, 6-3. Bolivia's Hugo Dellien, Serbia's Laslo Djere and Spain's Pedro Martinez also earned first-round victories.
Third-seeded Lorenzo Musetti (20) of Italy opposed Chilean qualifier Nicolas Jarry in the last match of the night, and Jarry pulled off a 6-4, 6-1 upset.
