One of the major problems impeding the sustainable development of goose production is low egg hatchability. Thus, it is imperative to develop more efficient ways to improve the hatching qualities of goose eggs. The present study aimed to investigate the effects of changes in the temperature regime of incubation on the availability of vitamins A (retinol) and E (α-tocopherol), as well as important elements of antioxidant protection in embryos and goslings. The initial heating of eggs at temperatures of 39 and 39.5 °C during the first 24 or 36 h was used as the main factor influencing goose embryo development. The dynamics in the content of antioxidant retinol and α-tocopherol were analyzed in the liver of embryos and goslings (up to 9 days of age) using the standard regime of incubation (at 38 °C) and the prior heating of eggs (at 39 °C and 39.5 °C) in the first 24 and 36 h of incubation. The obtained results provided new information about the effect of the initial heating of eggs on the function of antioxidants in the gosling’s body. It was confirmed that the age-related changes of retinol and α-tocopherol levels in goslings are similar to those found in chicks, ducklings, and turkey-poults. In addition, the effect of an egg’s weight on hatchability was established, with the hatchability of eggs in the medium weight class being higher than that of smaller and larger eggs. The efficiency of the redistribution of retinol from the yolk to the liver in day-old goslings was higher by 6% when exposing eggs to a temperature of 39.5 °C for 36 h. The initial heating regime at 39.5 °C led to an increased hatch of goslings, and to the evenness of their hatch from eggs of different weights, approaching the hatch values of the medium class. Overall, the α-tocopherol concentration in the liver of embryos and goslings, using the starting heating of eggs, was higher than that with the standard temperature mode of incubation. At the same time, the hatch rate of goslings increased by 9–13%, and the hatchability of eggs by 10–16%. For use in the practice of hatcheries and breeding poultry farms, it is recommended that goose eggs are heated for 36 h at 39.5 °C.