The main purpose of the present research was to evaluate the effect of varying levels of DA inclusion in maternal diet, in the form of powder, on the born-weaning growth performance (days 1–30) and intestinal health of neonatal rabbits. A total of 152 delivered maternal does (3.94 ± 0.05 kg) were allocated into four groups, with thirty-eight replicates of one doe each, and fed with a control diet (CON) supplemented with different levels of powdered DA (85 mg/kg (DA85), 170 mg/kg (DA170), and 340 mg/kg (DA340)) during lactation. The results show that dietary DA increased individual body weight at days 21 and 30 (p = 0.03 and p < 0.01) and showed a linear and quadratic effect on individual body weight and average daily gain (ADG) (p < 0.05), although ADG was not affected by DA. The serum progesterone (P) (p = 0.05) and glutathione peroxidase (GSH-Px) (p = 0.01) concentrations of maternal rabbits were increased in all maternal DA-supplemented groups and showed a linear and quadratic effect (p < 0.05), while a linear effect was observed in estrogen (E-2) (p < 0.05). Interestingly, the concentrations of the serum insulin-like growth factor-1 (IGF-I) (p < 0.01) and immunoglobulin (IgA and IgG) (p < 0.01 and p = 0.01) in neonatal rabbits were increased in all dietary DA groups, and both showed a linear and quadratic effect (p < 0.05). In addition, the serum interleukin (IL-1β) (p = 0.03) and catalase (CAT) (p = 0.04) concentrations were affected by DA supplements, but linear and quadratic effects were only observed in the catalase (CAT) of neonatal rabbits (p < 0.05). Importantly, the duodenal and jejunal villus heights (p < 0.04 and p = 0.01) and the ratio of villus height to crypt depth (p = 0.02 and p = 0.01) in neonatal rabbits were elevated in all DA-supplemented treatments, while a linear and quadratic effect was observed in jejunum, but a quadratic effect was observed in duodenum (p < 0.05). The ileal villus height (p < 0.01) was also elevated in all DA-supplemented treatments exhibiting both linear and quadratic effects (p < 0.05). Moreover, the expression levels of occludin (p = 0.04), zonula occludens-1 (ZO-1) (p < 0.01), IGF-1 (p = 0.03), and solute carrier family 5 member 9 (SCL5A9) (p < 0.01) in jejunum were markedly elevated in the DA175 and 340 mg/kg DA-supplemented groups and were showing linear and quadratic effects (p < 0.05). Finally, inflammation-related gene expression levels such as NF-κB, TLR4, and MYD88 in jejunal (p = 0.01, p = 0.04, and p < 0.01) and ileal (p = 0.04, p = 0.04, and p = 0.03) epithelium were decreased in the DA170 and DA340 groups (p < 0.05), while linear and quadratic effects were observed for TLR4 in jejunum and for NF-κB and TLR4 MYD88 in ileum (p < 0.05). In summary, as a functional additive, maternal DA supplementation with 170 and 340 mg/kg DA during lactation can promote the growth of neonatal rabbits, which is related to improved antioxidative capacity and immunity, as well as improved intestinal health in neonatal rabbits.