arXiv:1502.05820v1.-- et al. ; We present results on the spectral decomposition of 118 Spitzer Infrared Spectrograph (IRS) spectra from local active galactic nuclei (AGNs) using a large set of Spitzer/IRS spectra as templates. The templates are themselves IRS spectra from extreme cases where a single physical component (stellar, interstellar, or AGN) completely dominates the integrated mid-infrared emission. We show that a linear combination of one template for each physical component reproduces the observed IRS spectra of AGN hosts with unprecedented fidelity for a template fitting method with no need to model extinction separately. We use full probability distribution functions to estimate expectation values and uncertainties for observables, and find that the decomposition results are robust against degeneracies. Furthermore, we compare the AGN spectra derived from the spectral decomposition with sub-arcsecond resolution nuclear photometry and spectroscopy from ground-based observations. We find that the AGN component derived from the decomposition closely matches the nuclear spectrum with a 1σ dispersion of 0.12 dex in luminosity and typical uncertainties of ∼0.19 in the spectral index and ∼0.1 in the silicate strength. We conclude that the emission from the host galaxy can be reliably removed from the IRS spectra of AGNs. This allows for unbiased studies of the AGN emission in intermediate- and high-redshift galaxies - currently inaccesible to ground-based observations - with archival Spitzer/IRS data and in the future with the Mid-InfraRed Instrument of the James Webb Space Telescope. The decomposition code and templates are available at http://denebola.org/ahc/deblendIRS. ; A.H.-C. and A.A.-H. acknowledge funding by the Universidad de Cantabria Augusto González Linares programme and the Spanish Ministry of Economy and Competitiveness under grant AYA2012-31447, which is the party funded by the FEDER program. C.R.A. is supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme (PIEFGA-2012-327934). T.D.S was supported by ALPIA-CONICYI grant number 31130005. S.F.H. acknowledges support from the Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme (PIIF-GA-2013-623804). P.E. acknowledges support from the Spanish Programa Nacional de Astronomía y Astrofísica under grant AYA2012-31277. ; Peer Reviewed