Integrative analyses of genome-wide association studies (GWAS) and gene expression data across diverse tissues and cell types have enabled the identification of putative disease-critical tissues. However, co-regulation of genetic effects on gene expression across tissues makes it difficult to distinguish biologically causal tissues from tagging tissues. While previous work emphasized the potential of accounting for tissue co-regulation, tissue-specific disease effects have not previously been formally modeled. Here, we introduce a new method, tissue co-regulation score regression (TCSC), that disentangles causal tissues from tagging tissues and partitions disease heritability (or covariance) into tissue-specific components. TCSC leverages gene-disease association statistics across tissues from transcriptome-wide association studies (TWAS), which implicate both causal and tagging genes and tissues. TCSC regresses TWAS chi-square statistics (or products of z-scores) on tissue co-regulation scores reflecting correlations of predicted gene expression across genes and tissues. In simulations, TCSC powerfully distinguishes causal tissues from tagging tissues while controlling type I error. We applied TCSC to GWAS summary statistics for 78 diseases and complex traits (average N = 302K) and gene expression prediction models for 48 GTEx tissues. TCSC identified 27 causal tissue-trait pairs at 10% FDR, including well-established findings, biologically plausible novel findings (e.g. aorta artery and glaucoma), and increased specificity of known tissue-trait associations (e.g. subcutaneous adipose, but not visceral adipose, and HDL). TCSC also identified 30 causal tissue-trait covariance pairs at 10% FDR. For the positive genetic covariance between eosinophil count and white blood cell count, whole blood contributed positive covariance while LCLs contributed negative covariance; this suggests that genetic covariance may reflect distinct tissue-specific contributions. Overall, TCSC is a powerful method for distinguishing causal tissues from tagging tissues, improving our understanding of disease and complex trait biology.