The 20-year-old American dug deep to come through 4-6 6-4 6-2.

Spaniard Badosa, ranked 19th in the world but a former world number two, had looked in a good position when she served at 4-3 up in the second set.

But Gauff broke and rattled off the next four games to set up a decider, eventually wrapping up victory in two hours and 21 minutes.

She will face Karolina Muchova in Sunday’s decider after the Czech player defeated home favourite and fifth seed Qinwen Zheng 6-3 6-4.

Zheng, the Olympic champion, needed medical treatment for a hand injury after taking a tumble during the opening set.

Although both players had their opportunities, it was Muchova, who reached last year’s French Open final but is now ranked 49 in the world after wrist surgery, who took advantage.

The win saw her avenge defeat by Zheng in the final in Palermo in July and reach a second career WTA 1000 final after Cincinnati last year, where she lost to Gauff.

Badosa, who was as low as 140th in the rankings in May, was on a good run of form in Beijing which included victory over second seed and US Open finalist Jessica Pegula in the last 16.

And she took advantage of Gauff’s inconsistencies, particularly on her serve which produced 11 double faults, to win the opening set and break in the first game of the second.

Gauff is working with new coach Matt Daly after ending her partnership with Brad Gilbert last month following her fourth-round exit at the US Open, where she was defending her title.

She saved four break points to avoid going 4-1 down in the second set and from then the momentum was with the American, who regained her composure as Badosa dropped her first set of the week.

Gauff broke three times in the decider to seal the win to reach her first final since she triumphed in Auckland in January.

“I tried to stay focused, it was tough but mentally I just tried to reset,” she said.

Alcaraz, 21, continued his fine run of form against the Italian and has now won each of their last three encounters after coming through 6-7 (6-8) 6-4 7-6 (7-3).

It is the Spaniard’s fourth ATP Tour title this year and 16th of his career.

“Jannik once again showed he’s the best player in the world, he’s unbelievable and plays such a high level of tennis,” Alcaraz said.

“I had my chances in the first set and didn’t take it. In general, I’m proud of the way I dealt with the match and managed everything.

“I never lose hope but I know he has great stats. I knew I had to give everything I had to give myself the opportunity.”

Elsewhere, Great Britain’s Dan Evans was knocked out in the first round at the Shanghai Masters, having come through qualifying.

Evans, 34, forced a deciding set, which he led 4-2, but eventually lost out in a tie-break against Brazil’s Thiago Seyboth Wild, who won 6-2 4-6 7-6 (7-3) to set up a meeting with Russian world number five Daniil Medvedev.

Alcaraz gets better of Sinner in latest showdown

Since suffering a shock second-round exit at the US Open, Alcaraz has bounced back to win nine matches in a row, including helping Team Europe win the Laver Cup last month.

Alcaraz appeared to be on track for the perfect start when he moved into a 5-2 lead in the opening set, however, Sinner saved two set points to force a tie-break, which he went on to win.

The pivotal moment in the second arrived in the ninth game, when Alcaraz seized his opportunity to secure a break before serving out for the set to take it to a decider.

There has been little to separate the pair in the early stages of their careers, with their head-to-head record now 6-4 in Alcaraz’s favour, and it was another fine example of what the future of tennis has to offer in Beijing.

Sinner has made a habit of winning big games over the course of 2024, prevailing in all six of his previous finals – including the Australian Open and US Open – and he found an extra gear to mount another fightback in the final set.

Trailing 4-2 in the third, Sinner won three successive games to wrestle back momentum and send the encounter to a match-deciding tie-break.

Sinner looked firmly in control after opening up a 3-0 advantage, but Alcaraz would not be denied.

He rattled off seven successive points to claim victory and make it three wins on the bounce against the two-time Grand Slam champion.

“During the whole week I’ve been playing great tennis, some luck went to my side but I felt great on the court too,” Alcaraz said.

The win means he is the first player to win an ATP 500 title on all three surfaces.

‘System is not working well’ – Djokovic

Sinner was competing in Beijing after the World Anti-Doping Agency (Wada) announced on Saturday that it was appealing the decision to clear him of blame after he twice tested positive for a banned substance in March.

The 23-year-old returned positive for low levels of a metabolite of clostebol – a steroid that can be used to build muscle mass – during Indian Wells.

Wada said last month’s ruling by an independent tribunal to find Sinner had no fault or negligence was “not correct under the applicable rules”.

It has appealed to the Court of Arbitration for Sport (Cas) and said the player should be banned for “between one and two years”.

Sinner’s defence said he was inadvertently contaminated by the banned substance by his physiotherapist, Giacomo Naldi, and when Wada’s decision was announced, Sinner said he was “surprised” by the decision to appeal.

World number four Novak Djokovic, speaking at a news conference for the Shanghai Masters, called for the issue to be “resolved as soon as possible” and criticised the tennis anti-doping system’s “inconsistencies”.

“I think it’s quite obvious that we have a system that is not working well,” Djokovic said.

“There’s way too many inconsistencies, way too many governing bodies involved and this whole case is not helping our sport at all.

“Whatever is going to happen at the end of the day, just I wish for it to be resolved as soon as possible.”