Lancashire Police said a 34-year-old man is in custody and moved to reassure residents the incident is not terror related.
The Tesco Extra store, on Towngate, is expected to re-open at around 1.30pm on Thursday, August 24.
An eye-witness said customers were ushered to the rear of the premises while a number of panes of glass forming the shop's frontage were damaged by a man wielding a "lump hammer".
A police statement said: "An incident has occurred at Tesco Extra Towngate Leyland at 10.45am.
"Extensive damage was caused to numerous large windows at the front of the store, a male has been arrested.
"Police want to reassure the public that this was not a terror related incident."
The 34-year-old, from the Leyland area, has been arrested on suspicion of causing criminal damage, a force spokesman added.