Part of the answer to your question involves how you define "organ".   If you mean "a fully differentiated structural and functional unit in an animal that is specialized for some particular function"  that would probably be the placenta, which is actually extraembryonic and supplies developing embryo with nutrients from the mother. 

Next would be the heart; there is beating heart tissue by approximately three weeks after fertilization. 

The heart forms from mesodermal tissue in response to signals from the early embryonic endoderm. 

There are originally two regions of cardiogenic tissue; as development proceeds, these fuse to form the heart tube, which goes on to form the four-chambered human heart.

At around the same time, many other organs are beginning to form, including the gut, liver, pancreas, kidney and central nervous system.