As long as you have an approved H1B with a current start date, you may start work immediately. If your H1B starts October 1, you need to wait until October 1 to start. Good luck. You will want to visit a US embassy the first time you travel to have the H1 visa entered in your passport. This does involve an interview and is not automatic so see your immigration attorney before you travel outside the US.
You can only work for your H-1B petitioner once you have obtained H-1B status. You said that the petition was not filed as a change of status. The fact that the H-1B petition is approved is a separate issue from you receiving H-1B status. If you have not traveled outside the US and obtained in H-1B visa stamp, and entered the US in H-1B status as indicated on your new I-94, then you are not in H-1B status and you are not allowed to work (even though you have an approved petition). Either you need to depart the US, get an H-1B visa stamp, and enter the US in H-1B status, or your employer needs to file a new H-1B petition, this time seeking a change of status for you (assuming that you are currently in valid H-4 status), and you have to wait for the approval and new I-94 attached to the bottom of the notice until you can start working. If you are unsure what your status is, look at your I-94. If you have no I-94 that says H-1B, then you cannot work yet.
I agree with Mr. Wiedemann's answer below. The I-94 Arrival/Departure Document controls what a foreign national can do in the United States and for how long they can stay. If your I-94 indicates H-4 status you cannot work even with the USCIS H-1B approval notice unless it comes with a new tearout I-94 on the bottom of the notice.