There is nothing new except what has been forgotten
Numerous reviews, thesis introductions and Wikipedia articles on breath analysis state that modern breath analysis was founded by Linus Pauling, the famous chemist and one of the few people who won two Nobel prizes. Indeed, in his 1972 paper  he collected breath condensate and analyzed it by gas chromatography with a flame ionization detector.
Undoubtedly, it’s difficult to pinpoint “the start” of modern breath analysis. As a working hypothesis, we might say that the start was when volatile organic compounds in breath were instrumentally analyzed for the first time. From that perspective, modern breath analysis started more than two decades earlier and was pioneered by Arthur Jeffrey Dempster, a famous mass spectrometrist best known for the discovery of the uranium isotope 235-U.
In fact, two forgotten references from 1948 [2, 3] describe the mass spectrometric analysis of collected breath condensate and summarize that “it has been found … that much besides carbon dioxide is extracted from the blood and exhaled in the breath” . The acquired mass spectrum looks somewhat similar to the data we obtain nowadays in our lab: at virtually every nominal mass there is a peak observed, highlighting the complex composition of breath. Some components like acetone, ethanol and acetic acid were putatively assigned by the authors. Moreover, they conclude that “a systematic study of possible organic constituents … would, in all probability, make possible quantitative analyses of the normal and pathological volatile metabolism products” . Remarkably, this conclusion anticipated precisely what is done nowadays in the clinical trials of Zurich exhalomics, where the systematic and relative-quantitative study of volatile metabolites by mass spectrometry is used to find biomarkers. Likewise, nowadays studies of mice models on nutritional impact might have their ancestor in a 1951 paper , where exhaled acetone of rats with a certain diet was tracked by mass spectrometry.
It cannot be excluded that older references than the one from Dempster could pinpoint the start of modern breath analysis even earlier. For example, one year before Dempster, another publication  reported the on-line measurement of breath by mass spectrometry. Yet, it does not particularly mention organic metabolites and thus does not comply with our working hypothesis. Nonetheless, all the mentioned articles give evidence that modern breath analysis started at least two decades earlier than previously thought. Personally, as a mass spectrometrist, I was also excited to learn that it was mass spectrometry which paved the way into this thrilling research discipline.
 Teranishi, Mon, Robinson, Cary, Pauling, “Gas Chromatography of Volatiles from Breath and Urine”, Analytical Chemistry, 44(1), p. 18-20, 1972.
 Dempster, “Thirty years of mass spectroscopy [sic!]”, The Scientific Monthly, 67(3), p. 145-153, 1948.
 Dempster, Inghram, Hess, “The mass spectrometer analysis of volatile products of metabolism”, United States Atomic Energy Commission, Technical Information Division, Oak Ridge, Tennessee, 1948. Can be accessed via Google books.
 Pratt, Burr, Eden, Lorenz, “An application of mass spectrometer analysis to the study of respiratory metabolism”, Review of Scientific Instruments, 22(9), p. 694-696, 1951.
 Siri, “A mass spectroscope [sic!] for analysis in the low mass range”, Review of Scientific Instruments, 18(8), p. 540-545, 1947.