The Three Lions marched to a comprehensive 6-2 victory at the Khalifa International Stadium, but the game was marred by several stoppages for injuries.
Within the first few minutes of the match, Iran goalkeeper Alireza Beiranvand inexplicably tried to carry on after a sickening clash of faces with Morteza Pouraliganji, but he was eventually taken off.
England were then dealt two concerns of their own in the second half, as Maguire went down on the turf before gingerly walking off in the 70th minute to be replaced by Eric Dier.
Kane was also the victim of a poor challenge from Pouraliganji and appeared to awkwardly jar his knee in the turf, and the Three Lions captain remained down for a couple of minutes.
Kane was taken off for Callum Wilson with 14 minutes remaining, and Southgate was inevitably asked about the pair by reporters after the game.
The Three Lions boss alleviated fears over the pair's fitness, as he confirmed that Maguire's withdrawal was illness-related rather than concussion-related.
Meanwhile, Kane - who has been bedevilled by ankle problems in the past - was not an enforced change, and Southgate is confident that his captain has not sustained anything serious.
"Harry Maguire felt ill. (He) flagged it before the goal. There was no point carrying on but other than that no causes for concern," Southgate said, as quoted by The Athletic.
"Other than that haven't seen anything of concern (about injuries). I think Harry Kane is fine. It obviously looked like a bad tackle but he carried on in the game. We took him off really because we felt it was a moment we could do that."
Maguire was also asked by the publication if he felt OK after the game, to which the Manchester United centre-back replied "should be".
Southgate's decision to start Bukayo Saka in Monday's World Cup opener was justified, with the Arsenal man scoring twice and picking up the man of the match award.
Saka struck England's second goal after Jude Bellingham's header broke the deadlock, and Raheem Sterling was also on target before the half-time whistle blew.
Marcus Rashford and Jack Grealish came off the bench to add to England's tally in the second half alongside Saka, although Mehdi Taremi ended the game with a brace of his own.
Taremi's second came from the penalty spot in the 10th minute of second-half injury time, and Southgate - who set a new major tournament record of nine wins in charge of England - also admitted that his side's loss of concentration in the dying embers left him "fed up".
England will endeavour to make it two for two in the 2022 World Cup when they face the USA in their second Group B match on Friday evening.