An off-colour Alcaraz trailed midway through the opening set which was briefly halted by rain, but the Spaniard recovered to break Thompson's serve having squandered eight opportunities earlier and clawed his way back to 4-4 with a hold.
The reigning Flushing Meadows champion, who crashed in the Toronto quarter-finals last week, pounced again for a 6-5 lead before wrapping up the set but was under immediate pressure in the next as Thompson raced ahead 5-2.
Alcaraz won the next two games but his comeback fizzled out as he continued to struggle on his serve and forehand to allow world number 55 Thompson to force a third set.
An early break handed Alcaraz the advantage in the decider and the Spaniard slowly returned to his dominant best to claim his 50th tour level win of 2023 in a little over three hours.
"It wasn't an easy match but I found my best level at the end," Alcaraz said. "We were waiting around all day to play the match and when you step on court you have to start with the rain, so it wasn't easy.
"But you have to find a way to win these matches in this type of situation."
Earlier, title holder Borna Coric beat American Sebastian Korda 7-6(5) 6-4, while another home hope Taylor Fritz survived a first-set thriller to beat Czech Jiri Lehecka 7-6(14) 6-2.
Croatian Coric crashed out in Toronto in the opening round but produced a pristine display in Cincinnati with few unforced errors, wrapping up the match that began on Monday but was suspended due to bad weather.
Korda, who has suffered a handful of early exits since reaching the Queen's Club semi-finals, appeared to have the edge as he converted a breakpoint in the third game.
But 15th seed Coric levelled in the sixth and mounted a fine defence to rally from a 4-1 deficit in the tiebreak.
He won the crucial break in the third game of the second set, where he dropped only one first-serve point as Korda's level declined across the board, and pumped his fist as he closed the affair with an unreturnable serve.
"I was a little nervous coming into the match as I know I need to defend my title, so I put a little more expectation on myself," Coric said.
He next faces Poland's Hubert Hurkacz who beat Australian Thanasi Kokkinakis 7-6(2) 3-6 7-6(1).
Top American Fritz mustered all of his resources in an epic 30-point first set tiebreak with Lehecka.
The Monte Carlo semi-finalist saved five set points in the tense and closely fought opening set, where he produced fewer unforced errors to get the upper hand.
The momentum belonged to Fritz in the second set, though, when he never faced a break point and won all but one of his first-serve points.
Gael Monfils also advanced with a 3-6 6-4 6-3 victory over Briton Cameron Norrie after requiring a medical timeout late in the opening set to tend to his left calf.
"I tried to keep it very simple with my game. I think the ATP physio came on the court and did a great job, not only the treatment but he managed to secure my head," said Monfils.
"He said it would be fine and this meant the world for me."
Alejandro Davidovich Fokina beat Tomas Martin Etcheverry 6-3 6-3 to set up a clash with second seed Novak Djokovic who he had famously stunned en route to the Monte Carlo final last year.
Emil Ruusuvuori upset world number eight Andrey Rublev 7-6(10) 5-7 7-6(3) for his fourth top 10 victory while Felix Auger-Aliassime was denied back-to-back wins for the first time since Indian Wells in March as he fell 6-4 6-4 to Adrian Mannarino.
It was the end of the road for Washington champion Dan Evans who lost 6-4 6-3 to Lorenzo Musetti in their first-round meeting after Alexander Zverev eased past Grigor Dimitrov 6-2 6-2 in a battle between former champions.
