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.
Thunderstorms and heavy downpours of rain predicted to hit Lancashire as Met Office issues yellow weather warning
Man killed in late night crash on A6 Chorley Road near Blackrod
M6, M61 and M55 delays in Preston due to Blackpool Air Show and Lake District holiday traffic
Video shows man in van wielding weapon in Preston city centre fight
Preston man remains in a critical condition as Lancashire Police appeal for witnesses
"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.