The two women were hurt in the attack in the St James Street store in Burnley at 09:30am, Lancashire Police said.
A force spokesman said both were taken to hospital but their injuries were not believed to be life-threatening.
Supt Stasia Osiowy praised the “brave actions” of passers-by, who detained a 57-year-old man, who was then arrested on suspicion of attempted murder.
She said the attack was “not being treated as a terrorism incident, but due to some comments made at the scene, counter-terrorism detectives will be leading on the investigation”.
She said detectives were “keeping an open mind as to motivation”, but officers were “considering the mental health background of the man we have arrested”.
She added her thanks to “those members of the public who acted very quickly, and without regard to their own safety, in order to detain the attacker”.
“Without their brave actions, this incident, while serious, could have been so much worse.”
The force spokesman said the two women, a staff member in her 40s and a member of the public in her 60s, had “thankfully” not been seriously injured.
A knife was recovered at the scene and officers have stepped up patrols in the area.
Carl Stredder, who was shopping with his wife at the time, had stopped at the cash machine opposite Marks & Spencer when he heard shouting coming from the store.
He told the BBC he saw a man “holding down” another man before the emergency services arrived at the scene.
“Within a couple of minutes, the police had arrived. There were six or seven police cars, sirens all over the place,” he said.
“Then within another five or so minutes, an ambulance came. It was apparent by then that some sort of major incident had occurred.”
He added: “It was a bit of a frightener when we came home and realised in the cold light of day what could have happened