The former Under-21 manager was put in charge for World Cup qualifiers at home to Malta on Saturday and away to Slovenia three days later following the departure of Sam Allardyce after 67 days in charge.
Southgate's initial task is to steady the ship and build on the victory in Slovakia, Allardyce's only match as manager.
Admitting he was as surprised as anyone about Allardyce's downfall after he was secretly filmed making inappropriate comments during a Daily Telegraph probe into corruption, Southgate said his priority was results.
"We can't just have one leader, we need a team of leaders. That needs to emerge before we head to Russia (for the World Cup in 2018)," the former England defender and Middlesbrough manager told reporters.
"After a period of change over the summer and now this month, the most important thing at this time is leadership, both on and off the field, and Wayne (Rooney) has provided that over the last two years.
"But Wayne bears a big burden, too much at times."
Southgate named his first squad on Sunday with the only surprises being the recall of Stoke City full back Glen Johnson and the selection of Manchester United winger Jesse Lingard.
"In the past, we have gone young, young, young and there's only so long you can do that," Southgate said of his decision to bring back Johnson who last played for England in 2014.
"In key pressure moments there is a lack of big-match experience. To be able to bring a player in with 50-plus caps was something we thought long and hard about but felt it was the right call."
Southgate is prepared for the extra scrutiny he now faces and said taking over as England manager had been a lot easier than accepting the Middlesbrough job.
"If I was comparing it to Middlesbrough when I took over and everything was new, this has been far, far, calmer. I have been able to enjoy it a lot more," he said.
(Reporting by Martyn Herman, editing by Ed Osmond)
