(Abridged) We develop a novel technique to identify active galactic nuclei (AGNs) and study the nature of low-luminosity AGNs in the Sloan Digital Sky Survey. This is the first part of a series of papers and we develop a new, sensitive method to identify AGNs in this paper. An emission line luminosity in a spectrum is a sum of a star formation component and an AGN component (if present). We demonstrate that an accurate estimate of the star formation component can be achieved by fitting model spectra, generated with a recent stellar population synthesis code, to a continuum spectrum. By comparing the observed total line luminosity with that attributed to star formation, we can tell whether a galaxy host an AGN or not. We compare our method with the commonly used emission line diagnostics proposed by Baldwin et al. (1981; hereafter BPT). Our method recovers the same star formation/AGN classification as BPT for 85% of the strong emission line objects, which comprise 43% of our sample. A unique feature of our method is its sensitivity: it is applicable to 78% of the sample. We further make comparisons between our method and BPT using stacked spectra and selection in X-ray and radio wavelengths. We show that, while the method suffers from incompleteness and contamination as any AGN identification methods do, it is overall a sensitive method to identify AGNs. Another unique feature of the method is that it allows us to subtract emission line luminosity due to star formation and extract intrinsic AGN luminosity. We will make a full use of these features to study the nature of low-luminosity AGNs in Paper-II.