We present a new method to measure the top quark pair production cross section and the background rates with 2.7 fb$^{-1}$ of data from $p\bar{p}$ collisions at $\sqrt{s} =1.96$ TeV collected with the CDF II Detector. The size of the dataset was chosen to directly show the improvements of this new method. We select events with a single electron or muon, missing transverse energy, and at least one b-tagged jet. We perform a simultaneous fit to a jet flavor discriminant across nine samples defined by the number of jets and b-tags. We measure a top cross section of $\sigma_{t\bar{t}} = 7.64 \pm 0.57 \mathrm{(stat + syst)} \pm 0.45 \mathrm{(luminosity)}$ pb. An advantage of this approach is that many systematic uncertainties are measured in sit...