England fell to a home defeat by the Danes after Christian Eriksen’s expertly taken 35th minute penalty proved decisive.
Maguire was sent off for two bookable offences on Yussuf Poulsen and Kasper Dolberg in the 31st minute by Spanish referee Jesus Gil Manzano.
The 27-year-old’s dismissal capped what has been a miserable last few months for the former Leicester City defender who missed England’s September games after falling foul of the law while on holiday in Greece.
Southgate has backed Maguire to come through this sticky patch in his career.
"He's a top player. He's a massive part of what we do," Southgate told Sky Sports.
"He's having a period where he's having a lot of sticks thrown his way. He's big enough to come through that. He has our support and his club will be the same
"We have total belief in him. He's been getting all sorts for a while, from some people who should know better.
"None of us like to be in the centre of that sort of storm. When you're a bit older you can rationalise it and come to terms with it. He's experiencing it for the first time, although no player gets to the level he's been without having setbacks.
"His profile has changed because of the club he plays at and his success with us. He's got to keep going and the tide will turn."
England are now third in their group on seven points with only the group winners making it to the Nations League finals next summer.
