Southgate's side secured its second win in World Cup qualifying with first-half goals from Daniel Sturridge and Dele Alli to top its group after two games.
The caretaker boss was in charge for the first time in the wake of Sam Allardyce's departure
after just 67 days at the helm.
Allardyce, 61, was filmed by undercover Daily Telegraph reporters seemingly admitting that he knew of ways to skirt FIFA rules on player ownership.
He was summoned to a meeting at the Football Association and left "by mutual consent" after just one match, a 1-0 stoppage-time win against 10-man Slovakia in September.
Southgate, the England Under 21 manager, was handed the role for four games while the FA conducts a recruitment process for a full-time replacement.
Sturridge scored with a header from Jordan Henderson's cross after 29 minutes and Alli scrambled in a second before the break, but England was unable to build a bigger lead against a team ranked 176th
-- 164 places below the 1966 World Cup winner.
"The first half we played well, we could and should have had more goals but their goalkeeper made some good saves," the 46-year-old Southgate told UK match broadcaster ITV.
"Players subconsciously know the game is won. We could have made it more comfortable for ourselves.
"Given where we were 10 days ago I'm pleased with the results but there is always room for improvement."
Southgate also defended his captain Wayne Rooney, who was subjected to booing by a section of the crowd as it grew frustrated in the second half.
"I don't quite understand that (booing)" he told the post-match media conference.
"Every debate focuses on him. The onus on him is enormous. The criticism of him is, at times, unfair. But he plows on and plays with pride, captaining his country with pride."
England next travels to Slovenia for a testing third World Cup qualifier Tuesday and the focus will be on Southgate's selection and if Rooney will be picked in his more deep-lying midfield role.
Southgate won 57 caps for England between 1995 and 2004 and managed Middlesbrough from 2006-2009.