"We are not naive enough to think 36 points will keep us in the division," said Southgate, whose side have games against Sunderland, Portsmouth and Manchester City to come.
"There is still a lot of work to be done.
"We had an opportunity to wrap it up from that side of things and we haven't taken that.
"It was a big disappointment for us. The way we started we should have gone on and won it but we haven't, so we've got to go the extra yards in the next few weeks.
"However long it takes, we have to go that distance. Having been here as long as I have I should know that the opportunity to have three weeks of enjoyment was too much to ask for.
"We have to roll up the sleeves and dig into the well again, but we have got the character and the resilience to do that."