Context. After leaving the main sequence, massive stars undergo complex evolution, a process that is still poorly understood. With a population of hundreds of OB stars, the starburst cluster Westerlund 1 offers an unparallelled environment to study their evolutionary tracks. Aims. We characterise a large sample of evolved OB stars in the cluster, with the aim of determining cluster parameters and place stars in an evolutionary sequence. Methods. We used the FORS2 instrument on the VLT to obtain intermediate-resolution spectroscopy over the range 5800-9000 Å of about a hundred stars selected as likely members of the cluster based on their photometry. We developed criteria for their spectral classification using only spectral features in the range observed. We discuss these criteria, useful for spectral classification of early-type stars in the GAIA spectral region, in the appendix. Using these criteria, we obtain spectral classifications, probably accurate to one subtype, for 57 objects, most of which had no previous classification or a generic classification. Results. We identify more than 50 objects as OB supergiants. We find almost 30 luminous early-B supergiants and a number of less luminous late-O supergiants. In addition, we find a few mid B supergiants with very high luminosity, some of them displaying signs of heavy mass loss. All these stars form a sequence compatible with theoretical evolutionary tracks. In addition, two early B supergiants also show indication of heavy mass loss and may represent the evolutionary phase immediately prior to the Wolf-Rayet stage. We investigate cluster properties using the spectral types and existing photometry. We find that the reddening law to the cluster does not deviate strongly from standard, even though extinction is quite variable, with an average value A V=10.8. Though evolutionary tracks for high-mass stars are subject to large uncertainties, our data support an age of >5 Myr and a distance d5≈kpc for Westerlund 1. Conclusions. The spectral types observed are compatible with a single burst of star formation (the age range is very unlikely to be >1 Myr). Westerlund 1 shows its potentiality as a laboratory for massive star evolution, which can be fulfilled by detailed study of the population presented here. © 2010 ESO.