Norrie had been given a bye into the second round, where he beat Chun-Hsin Tseng of Taiwan in straight sets.
The world number 12, who reached the semi-finals of Wimbledon, recovered after losing his serve in the very first game to twice break Albot and take the opening set 6-3.
— ATP Tour (@atptour) August 5, 2022
Norrie then looked on course for another comfortable victory when forcing an early break in the second set to then lead 5-3.
However, Moldovan Albot, ranked 118 in the world, gained momentum after a hold and then a break to level at 5-5.
The 32-year-old then backed that up to move 6-5 ahead and pile the pressure back on Norrie’s faltering serve.
Norrie, though, then produced a big hold himself to send the match into a tie-break.
A short volley from Albot at the net was swiftly dispatched by Norrie to claim an early mini-break before moving 4-1 ahead and eventually closed out for a 6-3 7-6 (4) victory in just under two hours.
Speaking on court after his victory, Norrie said: “I know Radu, he is a great competitor and athlete. I didn’t really start my best again today so had to fight from behind.
“I was ahead in the second set and then got a little nervous there, then he upped his game.
“I served for the match, but played a terrible game. But that happens and I had enough confidence to accept it was OK and relax, then play a good tie-break after that.
“I am pleased to get through, get some rest and get ready for tomorrow.”
Let’s keep it going💪🏽 I love the energy
📸: Manuel Velasquez | Rodrigo Arangua pic.twitter.com/OPj9TQFeuB
— Félix AugerAliassime (@felixtennis) August 4, 2022
Norrie goes on to face second seed Felix Auger-Aliassime, who earlier beat American Steve Johnson 6-4 7-6 (3).
The Canadian has never lost against Norrie, holding a 4-0 head-to-head record.
“Every time I play Felix he plays really well and has beaten me every time so I am going to have to play well tomorrow,” Norrie said.
Elsewhere, world number one Daniil Medvedev coasted into the semi-finals with a 6-2 6-2 victory against Ricardas Berankis.
The Russian top seed will go on to play fourth seed Miomir Kecmanovic, who beat sixth seed Brandon Nakashima 6-2 6-4.