Yes, you can work in the U.S. with a K-1 visa, but you need to obtain a work permit called an Employment Authorization Document (EAD) before starting any employment. After entering the United States on a K-1 visa, the non-citizen fiancé(e) can apply for the EAD by filing Form I-765, Application for Employment Authorization, with the United States Citizenship and Immigration Services (USCIS).
It is important to note that the EAD granted to K-1 visa holders will be valid only for the 90-day duration of their K-1 visa status. After the marriage and filing of the I-485 application for adjustment of status, the non-citizen spouse can apply for a new EAD based on their pending adjustment of status application. The new EAD will have a longer validity, typically one year, and can be renewed until the foreign spouse receives their green card.