Neonatal calf diarrhoea (NCD) poses a significant health challenge in cattle herds, resulting in considerable economic losses and antimicrobial use. In response to the escalating threat of antimicrobial resistance, viable alternatives are imperative, aligning with European policies. This study evaluated the in-milk supplementation of the chestnut and quebracho tannin extract in preweaning calves on performance, diarrhoea occurrence, Cryptosporidium spp. shedding, protein digestibility, and intestinal health. Twenty newborn calves were divided, after colostrum administration, into two experimental groups for 30 days as follows: the control (CTRL) was fed with whole milk and solid feed, and tannins (TAN) were fed whole milk supplemented with 6/g day of tannin extract and solid feed. Faecal samples were collected on days 0, 3, 7, 14, and 30 for the evaluation of Cryptosporidium oocyst shedding and protein digestibility. Faecal consistency was evaluated during the sampling using the faecal score scale (0–3 scale, considering diarrhoea > 1). The results showed a significant reduction in diarrhoea frequency in the TAN compared to the CTRL group (p < 0.05) over 30 days of the trial. The prevalence of Cryptosporidium spp. was generally low (12%), considering all analysed samples. Protein digestibility revealed comparable values for the TAN and CTRL groups, suggesting that tannins did not negatively affect milk protein availability. In conclusion, the in-milk supplementation of 6/g day of the chestnut and quebracho tannin extract could be considered a valuable functional feed additive to decrease NCD occurrence, thus supporting animal health and decreasing antibiotic use in livestock.