I know Berkeley considers Introductory Stats to count towards applicant’s “core” preparation despite not being a transferable major requirement since it a relevant subject.
Looking at UCSB’s degree requirements ( https://my.sa.ucsb.edu/Catalog/2011-2012/Documents/2011_Majors/LS/Econ/Econ-BA-2011.pdf ) shows that PSTAT-5E, which is the direct equivalent of most introductory stats courses, is under “preparation for the major” so I don’t see why it wouldn’t.
I would verify with UCSB Economics directly to be completely certain.