The galactic cluster Westerlund 1 contains a rich population of evolved, massive stars, and a high binary fraction has been inferred from previous multiwavelength observations. We use multi-epoch spectroscopy of a large sample of early-type stars to identify new binaries and binary candidates in the cluster. VLT/FLAMES was used to obtain spectra of ~100 OB stars over a 14-month baseline in 2008 and 2009, supplemented with follow-up observations in 2011 and 2013, and we identify 20 new OB I--III binaries, a WN9h binary, and a WC9d binary, greatly increasing the number of directly confirmed binary systems in Westerlund 1, while 12 O9--9.5 Iab--III stars are identified as candidate binaries. The 173.9 day SB1 W1030 represents the first longer-period system identified in the cluster, while the determination of a 53.95 day period for W44/L makes it the first Wolf-Rayet binary in Westerlund 1 with a confirmed orbital period greater than ten days. Our results suggest the binary fraction in the OB population is at least 40%, and may be significantly higher. (Abridged)