The 20-year-old Liverpool defender was sent off in the second half of the European Championship qualifier in Tampere for kicking Tim Vayrynen.
The incident in the 1-1 draw saw Wisdom handed a one-match ban, which UEFA extended by a further three games for punching Dani Hatakka in the chest as he left the pitch.
The Reds defender will as a result miss the upcoming qualifiers at San Marino and against Lithuania, as well as next month's clashes with Finland and San Marino.
Southgate revealed he went to Liverpool to see Wisdom after the incident and expects the whole team to learn from his mistake.
"I went to see him because I think it was important that he realised we are supportive of him," Southgate said, speaking at St George's Park.
"But for all of us in the squad, it was a lesson to learn that in European football there are certain things [you can't do]. Games are refereed differently and keeping control is important.
"Andre suffered a harsh lesson because it is a four-game ban and is one which we as a squad must take on board."
Southgate confirmed that Wisdom will be replaced as captain by Under-21s veteran Jack Butland for the upcoming matches.
With 14 caps for the Young Lions and experience with the both senior team and Team GB, the on-loan Barnsley goalkeeper is seen as the best candidate to fill the void.
"Jack Butland will be captain," Southgate said. "First and foremost his professionalism and his standing with the group is very high.
"He is our most experienced player, he has obviously played for the seniors, and around the squad I think he is a very good influence.
"He connects well with all of the squad and I think that is important."
Butland is not expected to have much to do against San Marino on Thursday, which is fortunate given full-backs Luke Shaw and Jack Robinson will miss out through illness.
The pair could feature against Lithuania next week, although Derby's Will Hughes will be unavailable having withdrawn from the squad through injury.
Other notable absentees from the Young Lions squad are Swansea's Jonjo Shelvey and Sunderland striker Connor Wickham.
The pair played a part in each of England's three miserable performances at the European Championship in Israel and are not currently part of Southgate's plans.
"I know enough about Jonjo," he said. "I wanted to look at some different players.
"There is a fair chance that if we went to a tournament in two years' time that he might be involved at a more senior level.
"Also he has had a lot of matches going back to pre-season, when Swansea were in Europe and involved in their first-team.
"With midfield and out wide, it is dangerous to say but I have an embarrassment of riches, really.
"Some of the younger ones I want to have a look at because I think they are more likely to be involved going towards a tournament and I would like to bed them in now.
"Connor was not able to report last time and hasn't really played any football since. It is as simple as that, really.
"Where Danny Ings scoring regularly at Burnley, has been in outstanding form, and Nick Powell at Wigan the same.
"Danny is the addition, if you like, and I just felt his form justified that, but we know about Connor from seeing him play in the younger age groups.
"But at the moment, he has not really been able to play first-team football or anything underneath that."