A partial-wave analysis of the decay J/psi -> K+ K- pi(0) has been made using (223.7 +/- 1.4) x 10(6) J/psi events collected with the BESIII detector in 2009. The analysis, which is performed within the isobar-model approach, reveals contributions from K-2* (1430)(+/-), K-2* (1980)(+/-) and K-4* (2045)(+/-) decaying to K-+/-pi(0). The two latter states are observed in J/psi decays for the first time. Two resonance signals decaying to K+ K- are also observed. These contributions cannot be reliably identified and their possible interpretations are discussed. The measured branching fraction B(J/psi -> K+ K- pi(0)) of (2.88 +/- 0.01 +/- 0.12) x 10(-3) is more precise than previous results. Branching fractions for the reported contributions are presented as well. The results of the partial-wave analysis differ significantly from those previously obtained by BESII and BABAR.