Measurements of CP asymmetry in charmless B→PV decays are presented, where P and V denote a pseudoscalar and a vector meson, respectively. Five different B→PV decays from four final states, B±→π±π+π-, B±→K±π+π-, B±→K±K+K- and B±→π±K+K- are analyzed. The measurements are based on a method that does not require full amplitude analyses, and are performed using proton-proton collision data at a center-of-mass energy of 13 TeV collected by LHCb between 2015 and 2018, corresponding to an integrated luminosity of 5.9 fb-1. In the π+π- P-wave, in the region dominated by the B±→ρ(770)0K± decay, a CP asymmetry of ACP=+0.150±0.019±0.011 is measured, where the first uncertainty is statistical and the second is systematic. This is the first observation of CP violation in this process. For the other four decay channels, in regions dominated by the B±→ρ(770)0π±, B±→K(¯) ∗(892)0π±, B±→K(¯) ∗(892)0K± and B±→φ(1020)K± decays, CP asymmetries in the P-wave compatible with zero are measured.