The Three Lions ran riot at the Khalifa International Stadium, with Bukayo Saka, Jude Bellingham and Raheem Sterling all on target in the first half of the contest.
Saka doubled his tally in the second half before substitutes Marcus Rashford and Jack Grealish got in on the act, but Mehdi Taremi managed to net two consolations for Team Melli.
Taremi's second of the day came from the penalty spot in the 10th minute of second-half injury time, and Southgate was not overjoyed at his side's loss of concentration near the end.
Asked by BBC Sport for his immediate feelings after the win, Southgate said: "A bit fed up with the end of it really! To win by that margin and play as we did, you've got to be really happy.
"It was a sticky first half, lots of stoppages but we were a real threat throughout - our pressing was really good, our movement was really good.
"I've got to be really happy but we shouldn't be conceding two goals at that stage of the match and we'll have to be right on our game against the [United] States."
The first half was halted early on as Iran goalkeeper Alireza Beiranvand was involved in a nasty collision with teammate Majid Hosseini, and he controversially tried to carry on before being substituted.
Ten minutes were also added on at the end of the second half following a head injury to Harry Maguire, who had put in a strong showing amid swathes of criticism before being taken off.
Southgate admitted that the lengthy periods of stoppage time affected his side's concentration, and he has called for improvement from his players ahead of the rest of the tournament.
"To win by that margin - to play as we did for the majority of the game - we've got to be really happy. That's a great marker to put down," Southgate added. "It's a great start but we're going to have to be better.
"We had, I think, 24 minutes of added time across the game, so it's a long time to focus - but we just lost concentration and when we play at a slower tempo, we're nowhere near as effective.
"I'm a miserable so-and-so and I should be more excited, but I've got to keep the team on the right track and we will have to be better."
Southgate also revealed that Maguire's substitution was illness-related rather than due to fears of a concussion, while captain Harry Kane appears to have avoided serious injury after awkwardly jarring his leg in the second half.
England will aim to make it two for two in Group B when they take on the USA on Friday before rounding off their group stage versus Wales on November 29.