Type 2A DNA topoisomerases (Topo2A) remodel DNA topology during replication, transcription and chromosome segregation. These multisubunit enzymes catalyze the transport of a double-stranded DNA through a transient break formed in another duplex. The bacterial DNA gyrase, a target for broad-spectrum antibiotics, is the sole Topo2A enzyme able to introduce negative supercoils. We reveal here for the first time the architecture of the full-length Thermus thermophilus DNA gyrase alone and in a cleavage complex with a 155 bp DNA duplex in the presence of the antibiotic ciprofloxacin, using cryo-electron microscopy. The structural organization of the subunits of the full-length DNA gyrase points to a central role of the ATPase domain acting like a 'crossover trap' that may help to sequester the DNA positive crossover before strand passage. Our structural data unveil how DNA is asymmetrically wrapped around the gyrase-specific C-terminal β-pinwheel domains and guided to introduce negative supercoils through cooperativity between the ATPase and β-pinwheel domains. The overall conformation of the drug-induced DNA binding-cleavage complex also suggests that ciprofloxacin traps a DNA pre-transport conformation.