England were a goal down to Slovakia and on the verge of an embarrassing last-16 exit in Gelsenkirchen when Bellingham saved Southgate and his team-mates with a superb bicycle kick in the final seconds.
Harry Kane then popped up early in extra-time with a trademark header to give under-par England a hard-fought victory.
"We knew that one goal would change the whole atmosphere in the ground. I believed that we could get that goal but I didn't think it would be as late as it was," Southgate told reporters.
"It was a night where I wasn't prepared, I wasn't ready to go home and clearly the players felt the same."
Southgate praised Kane and Bellingham, who have scored all four of England's goals between them, for winning the match despite showing clear signs of exhaustion.
"They're two players who you could have arguably taken off with fifteen minutes to go, they're out on their feet physically," added Southgate.
"But that's what they're capable of and that's why we stuck with them."
England will face Switzerland in the quarter-finals in Duesseldorf next weekend after turning Sunday's match around at the last.