Alcaraz arrived at Flushing Meadows as one of the tournament's major question marks, having not played since April due to a wrist injury, but he had his best weapon working for him as he struck 20 forehand winners in a 6-4 6-4 6-4 victory.
"You don't know what's going to happen in the match after four-and-a-half months almost. You don't know what's going to happen in your body, in your mind, everything. It's totally different," said the 23-year-old, who next plays Portugal's Jaime Faria.
"But I just tried to play one game at a time and tried to feel good, to enjoy playing points, feeling the love from the people. Hopefully they enjoyed it.
"It's been a really tough five months for me. I missed it - I missed the crowd, the energy, I missed competition, I missed everything."
Alcaraz admitted he had suffered "more nerves than usual".
"I don't know how far I'm going to go in the tournament. I didn't know if I was going to lose this one, if I'm going to make the second round, final," he said.
"But what I just really want to do is enjoy every time that I'm going to walk into the court feeling the energy from the people, feel the competition."
Enthusiastically welcomed onto Arthur Ashe Stadium, Alcaraz whacked a forehand winner out of Safiullin's reach for the break in the seventh game and looked his usual self as the Russian was unable to tame his opponent's lethal serve on set point.
The Spaniard went down an early break in the second set but Safiullin gave up the momentum immediately as he sent a volley out on break point in the third game.
Alcaraz painted the line to go up another break in the seventh and was in total control in the third set, where he converted a break point with his powerful forehand.
Another top-10 seed followed Novak Djokovic out when No.10 Arthur Fils, coming off the title in Cincinnati, was ousted by Stefanos Tsitsipas 4-6, 7-6 (7-3), 6-1, 6-4.
Tsitsipas, who is unseeded at the US Open for the first time, said ending a coaching relationship with his father Apostolos had brought much-needed calm to his tennis life and rebuilt his confidence.
"I realised that having people who sometimes bring the worst out of you instead of the best ... these are some of the things that need fixing," the 28-year-old Greek said.
"As much as I love the person that I work with, my father, and we have an amazing relationship outside of the tennis court, these sort of dynamics don't serve my purpose anymore as much."
Switzerland's Stan Wawrinka, the 2016 champion, fell to Italy's Matteo Berrettini 7-6(4) 7-6(3) 6-0. The 41-year-old triple grand slam winner bid goodbye to the tournament before his retirement at the end of the season.
"I'm going to miss it for sure," Wawrinka said.
Home hope Ben Shelton overcame Dutchman Tallon Griekspoor 1-6 6-1 7-6(3) 6-2 before Frances Tiafoe prevailed 6-4 4-6 3-6 6-3 6-4 in an all-American clash with Martin Damm.
No.14 seed Learner Tien overcame Portugal's Nuno Borges 6-4 6-1 6-2, while his fellow American Brandon Nakashima, the 16th seed, crushed Argentina's Sebastian Baez 6-2 6-1 6-0.
Other men's winners on Monday included Czech 17th seed Jakub Mensik, Italian 21st seed Luciano Darderi, and Chilean 25th seed Alejandro Tabilo.
