Focal oncogene amplification and rearrangements drive tumor growth and evolution in multiple cancer types. We present AmpliconArchitect (AA), a tool to reconstruct the fine structure of focally amplified regions using whole genome sequencing (WGS) and validate it extensively on multiple simulated and real datasets, across a wide range of coverage and copy numbers. Analysis of AA-reconstructed amplicons in a pan-cancer dataset reveals many novel properties of copy number amplifications in cancer. These findings support a model in which focal amplifications arise due to the formation and replication of extrachromosomal DNA. Applying AA to 68 viral-mediated cancer samples, we identify a large fraction of amplicons with specific structural signatures suggestive of hybrid, human-viral extrachromosomal DNA. AA reconstruction, integrated with metaphase fluorescence in situ hybridization (FISH) and PacBio sequencing on the cell-line UPCI:SCC090 confirm the extrachromosomal origin and fine structure of a Forkhead box E1 (FOXE1)-containing hybrid amplicon.