TMZ has obtained surveillance video of 3 burglars who entered an apartment unit in Josh's building earlier this month and made off with $7,000 in jewelry and other items. It is impossible to identify the faces of any of the 3 men, but the victim and a witness in the building told police they were "100% positive" it was Josh based on his jawline and the way he moved.

Turns out ... the "Desperate Housewives" star had an airtight alibi. He came home late afternoon and never left his apartment for the entire evening. We know that because there's surveillance video of his front door and that's the only way to enter and exit his unit.

The victim called police several times later that night and said he was scared to death Josh would come into his apartment and hurt him. The neighbor had 4 guns and told cops the burglars didn't take any of them and believed that was further evidence Josh was the culprit because he was afraid of guns.

Law enforcement sources tell TMZ, because the victim was afraid, they took the highly unusual step of going to the apartment building at around midnight and arresting Josh for burglary. They had not looked at the other video that showed he had an alibi.

Law enforcement sources say Josh's reaction was telling. He said, "How could I have committed a burglary?" rather than saying, "I didn't do it."

In the end ... the video clearly shows one thing -- the person who the victim identified as Josh was much younger than the actor.