H1 visa is the stamp you got on your passport. That's something you need to enter the States.
Not everyone with H1 status has an H1 visa. For example, one changed his status from F1 to H1, but he didn't leave U.S., then a valid H1 visa is not necessary. But currently he is in H1 status (determined by the I797 he got from the INS)
Visa is something you need only when you want to enter U.S. You don't need it for your legal staying