We report the results of the observation of the nearby satellite galaxy Segue 1 performed by the MAGIC-I ground-based gamma-ray telescope between November 2008 and March 2009 for a total of 43.2 hours. No significant gamma-ray emission was found above the background. Differential upper limits on the gamma-ray flux are derived assuming various power-law slopes for the possible emission spectrum. Integral upper limits are also calculated for several power-law spectra and for different energy thresholds. The values are of the order of 10-11 ph cm-2 s-1 above 100 GeV and 10-12 ph cm-2 s-1 above 200 GeV. Segue 1 is currently considered one of the most interesting targets for indirect dark matter searches. In these terms, the upper limits have been also interpreted in the context of annihilating dark matter particles. For such purpose, we performed a grid scan over a reasonable portion of the parameter space for the minimal SuperGravity model and computed the flux upper limit for each point separately, taking fully into account the peculiar spectral features of each model. We found that in order to match the experimental upper limits with the model predictions, a minimum flux boost of 103 is required, and that the upper limits are quite dependent on the shape of the gamma-ray energy spectrum predicted by each specific model. Finally we compared the upper limits with the predictions of some dark matter models able to explain the PAMELA rise in the positron ratio, finding that Segue 1 data are in tension with the dark matter explanation of the PAMELA spectrum in the case of a dark matter candidate annihilating into τ+τ-. A complete exclusion however is not possible due to the uncertainties in the Segue 1 astrophysical factor.