Recently, Fe–based metal–organic frameworks (Fe–MOFs) have been widely investigated as Fenton-like catalysts for environmental applications. However, their catalytic performance is limited by the insufficient number of Lewis acid sites for hydrogen peroxide (H2O2) activation and the slow Fe(III)–Fe(II) conversion. To overcome these limitations, coordinatively unsaturated iron–copper bimetallic sites may serve as Lewis acid sites, thus enhancing Fenton-like catalytic activity. Herein, we successfully synthesized a new Fenton-like catalyst of CUMSs/MIL-101(Fe,Cu), i.e., MIL-101 (MIL, Matérial Institute Lavoisier) with mixed valence of Fe(II)/Fe(III) and Cu(I)/Cu(II) as coordinatively unsaturated metal sites (CUMSs). Further, we evaluated the performance of catalytically activating H2O2 for ciprofloxacin (CIP) degradation. The results indicate a rapid CIP degradation by dosing CUMSs/MIL-101(Fe,Cu) (0.1 g·L−1) and H2O2 (3 mM) in the neutral pH range, with a 20-fold higher apparent rate constant than in a MIL-101(Fe)/H2O2 system. Various conditions of initial pH, catalyst amount, H2O2 concentration, and coexisting anions on the CIP degradation were systematically investigated for optimizing CIP degradation. Furthermore, we identified the intermediates during the CIP degradation and proposed the possible degradation pathways. We concluded that the thermodynamically favorable reaction between Cu(I) and Fe(III) and π–cation interactions accelerate the redox cycles of Fe(II)/Fe(III) and Cu(I)/Cu(II) CUMSs and this effect improves the Fenton-like performance over a broad pH range. Density functional theory calculations further clarified the dissociation process and the corresponding energy barriers of H2O2 at the atomic level. This work offers a new way for designing highly efficient, stable and harmless Fenton-like catalysts for achieving excellent environmental remediation efficiency over a wide pH range.