We report the discovery of non-thermal pulsed X-ray/soft \gamma -ray emission up to \sim 150 keV from the anomalous X-ray pulsar AXP 1E 1841-045 located near the centre of supernova remnant Kes 73 using RXTE PCA and HEXTE data . The morphology of the double-peaked pulse profile changes rapidly with energy from 2 keV up to \sim 8 keV , above which the pulse shape remains more or less stable . The pulsed spectrum is very hard , its shape above 10 keV can be described well by a power law with a photon index of 0.94 \pm 0.16 . 1E 1841-045 is the first AXP for which such very-hard pulsed emission has been detected , which points to an origin in the magnetosphere of a magnetar . We have also derived the total emission spectrum from the Kes73/1E 1841-045 complex for the \sim 2-270 keV energy range using RXTE HEXTE and XMM-Newton PN data . A comparison of the total emission from the complex with the pulsed+DC emission from 1E 1841-045 as derived from Chandra ACIS CC-mode data ( ) leaves little room for emission from Kes 73 at energies near 7 keV or above . This suggests that the HEXTE spectrum above \sim 10 keV , satisfactorily described by a power law with index 1.47 \pm 0.05 , is dominated by emission from 1E 1841-045 . In that case the pulsed fraction for energies above 10 keV would increase from about 25 % near 10 keV to 100 % near 100 keV . The origin of this DC-component extending up to \sim 100 keV is probably magnetospheric and could be a manifestation of pulsed emission which is “ on ” for all phases .