USB and eSATA use very different protocols to exchange data between devices. So NO, there is no simple adapter to convert one to the other. Interconnection would require hardware to translate one system to the other and backwards. Normally that is done by a mobo in a computer that has the required port management chips and software to convey data between the two.
You say your new external drive "has the new keyed USB3 connector". I presume you mean the cable that came with it has that type of MALE connector on the end that plugs into your laptop. However, your laptop does not have a matching USB3 FEMALE socket. It does have an eSATA socket. I would bet that the laptop also has a USB2 socket. Now, look again at the cable with the...