The 34-year-old, nominated for best actor at the awards for his role in The Danish Girl, made the announcement during an interview on the red carpet.
When American TV host Ryan Seacrest asked if there was a baby on the way, Redmayne replied: "That is true!".
After Redmayne confirmed the news, Seacrest presented the dad-to-be with two babygros - one with the words 'you should see my crib' and another which read 'milking my 15 minutes'.
"Thank you so much," Redmayne said as his wife laughed and said the gifts were "amazing".
A source told The Sun at the time: "Eddie and Hannah are delighted and extremely excited.
"They don’t want to find out the sex of the baby but are happy to tell close friends and family they’re expecting."
The baby is reportedly due June 2016.
Redmayne married Bagshawe in December 2014 in a small ceremony after dating for two years.