Tandem mass (MS/MS) spectrometry has become the method of choice for protein identification and has launched a quest for the identification of every translated protein and peptide. However, computational developments have lagged behind the pace of modern data acquisition protocols and have become a major bottleneck in proteomics analysis of complex samples. As it stands today, attempts to identify MS/MS spectra against large databases (e.g., the human microbiome or 6-frame translation of the human genome) face a search space that is 10-100 times larger than the human proteome, where it becomes increasingly challenging to separate between true and false peptide matches. As a result, the sensitivity of current state-of-the-art database search methods drops by nearly 38% to such low identification rates that almost 90% of all MS/MS spectra are left as unidentified. We address this problem by extending the generating function approach to rigorously compute the joint spectral probability of multiple spectra being matched to peptides with overlapping sequences, thus enabling the confident assignment of higher significance to overlapping peptide-spectrum matches (PSMs). We find that these joint spectral probabilities can be several orders of magnitude more significant than individual PSMs, even in the ideal case when perfect separation between signal and noise peaks could be achieved per individual MS/MS spectrum. After benchmarking this approach on a typical lysate MS/MS dataset, we show that the proposed intersecting spectral probabilities for spectra from overlapping peptides improve peptide identification by 30-62%.