The extent to which gene fusions function as drivers of cancer remains a critical open question. Current algorithms do not sufficiently identify false-positive fusions arising during library preparation, sequencing, and alignment. Here, we introduce a new algorithm, DEEPEST, that uses statistical modeling to minimize false-positives while increasing the sensitivity of fusion detection. In 9,946 tumor RNA-sequencing datasets from The Cancer Genome Atlas (TCGA) across 33 tumor types, DEEPEST identifies 31,007 fusions, 30% more than identified by other methods, while calling ten-fold fewer false positive fusions in non-transformed human tissues. We leverage the increased precision of DEEPEST to discover new cancer biology. For example, 888 new candidate oncogenes are identified based on over-representation in DEEPEST calls, and 1,078 previously unreported fusions involving long intergenic noncoding RNAs partners, demonstrating a previously unappreciated prevalence and potential for function. Specific protein domains are enriched in DEEPEST calls, demonstrating a global selection for fusion functionality: kinase domains are nearly 2-fold more enriched in DEEPEST calls than expected by chance, as are domains involved in (anaerobic) metabolism and DNA binding. DEEPEST also reveals a high enrichment for fusions involving known and novel oncogenes in diseases including ovarian cancer, which has had minimal treatment advances in recent decades, finding that more than 50% of tumors harbor gene fusions predicted to be oncogenic. The statistical algorithms, population-level analytic framework, and the biological conclusions of DEEPEST call for increased attention to gene fusions as drivers of cancer and for future research into using fusions for targeted therapy.