Norrie booked his place in the last four by upsetting young gun Alcaraz 7-6 (7-4) 6-7 (4-7) 6-4 in a titanic tussle that lasted more than three hours, after the Spanish star mounted an incredible fightback from a set and 4-1 down.
Norrie looked to be storming towards a straigt sets win before Alcaraz came roaring back to force a third set, though the Briton was ultimately able to hold him off by winning the points that mattered, saving 11 of the 13 break points he faced on the night.
"That was unbelievable," Norrie said after beating the third-seeded Spaniard. "Credit to Carlos. I was up a set and a break and I kind of lost a little bit of vision. I was thinking a little bit too much about the finish line rather than focussing on how I was winning points and I honestly got a little bit tight and he raised his level, didn't give me anything.
"I just wanted to hang tough with him and I think the only place I had him better was the legs and the physicality," he added. "So I was just trying to make every rally as physical as I could and make it tough for him to finish points."
Norrie moves on to face Borna Coric, who advanced after dispatching Felix-Auger Aliassime 6-4 6-4, having already accounted for the scalp of Rafael Nadal in what is turning out to be a great week for the Croatian.
Coric moves on to his first ATP Masters 1000 final since 2019, while the world number 152 is the second-lowest-ranked Cincinnati semi-finalist since the start of the ATP Rankings in 1973.
"I just came [to Cincinnati] and I just started to serve very good," he said. "I don't want to say too good because then tomorrow I'm not going to serve good, so I'm going to just say it was very solid. I was serving a lot last week. I took that risk with my shoulder. I can never know how much I can serve but I just took the risk and I was practising really hard."
The second semi-final is another blockbuster affair between world number one Daniil Medvedev and Stefanos Tsitsipas.
Medvedev, who will be the top seed at the US Open later this month, recovered from a slow start to dispatch American Taylor Fritz 7-6 (7-1) 6-3 to book his place in the last four on Friday.
"I'm happy with my level," said the 26-year-old. "If I can continue to play this way, and especially during the match raise my level, I can beat anybody."
Tsitsipas, meanwhile, survived a scare against the big-serving John Isner in a rain-interrupted clash that eventually went his way 7-6 (7-5) 5-7 6-3. Isner had Tsitsipas under all sorts of pressure in the first set and picked up a break late in the second set to force a decider, but the Greek star was ultimately able to prevail.
"He's one of the best," said fourth seed Tsitsipas of Medvedev. "I'm going to have to play my game. Of course, an obstacle like Daniil is never easy, but I'm going to try to stay concentrated, try to approach my game in the most precise, best manner. I'll let my tennis do the talking, and the rest will present itself if it's something that I deserve."