This paper presents the first results from a new citizen science project: Galaxy Zoo Supernovae. This proof-of-concept project uses members of the public to identify supernova candidates from the latest generation of wide-field imaging transient surveys. We describe the Galaxy Zoo Supernovae operations and scoring model, and demonstrate the effectiveness of this novel method using imaging data and transients from the Palomar Transient Factory (PTF). We examine the results collected over the period 2010 April-July, during which nearly 14 000 supernova candidates from the PTF were classified by more than 2500 individuals within a few hours of data collection. We compare the transients selected by the citizen scientists to those identified by experienced PTF scanners and find the agreement to be remarkable - Galaxy Zoo Supernovae performs comparably to the PTF scanners and identified as transients 93 per cent of the ̃130 spectroscopically confirmed supernovae (SNe) that the PTF located during the trial period (with no false positive identifications). Further analysis shows that only a small fraction of the lowest signal-to-noise ratio detections (r > 19.5) are given low scores: Galaxy Zoo Supernovae correctly identifies all SNe with ≥8σ detections in the PTF imaging data. The Galaxy Zoo Supernovae project has direct applicability to future transient searches, such as the Large Synoptic Survey Telescope, by both rapidly identifying candidate transient events and via the training and improvement of existing machine classifier algorithms. This publication has been made possible by the participation of more than 10 000 volunteers in the Galaxy Zoo Supernovae project ().