The Three Lions were held 1-1 at Wembley on Wednesday (AEDT), ending a run of 21 consecutive home qualifying wins.
They had to come from behind, too, with Roland Sallai dispatching a penalty after Luke Shaw was penalised for a high boot, before John Stones nodded a leveller.
Southgate sought to change the game – avoiding a repeat of the Poland game, where he made no substitutions in an England match for the first time since Euro 96 – but Jack Grealish was an odd choice for the first withdrawal.
Grealish had caused Hungary problems, unlike Kane and Sterling, who both followed.
Although Sterling had two of England's biggest chances, he could not beat Peter Gulacsi, while Kane's run of goals in 15 consecutive qualifiers came to an end without the struggling Tottenham superstar netting.
Southgate was asked if poor club form, with Sterling out of the first team at Manchester City, had contributed to below-par showings from two of England's most consistent performers.
But he told ITV: "I don't think we should look at individuals, because collectively we didn't perform at the level we needed to.
"When you have that sort of situation, it's difficult for individuals as well."
Southgate took a similar tact when asked about the make-up of his midfield, where Foden again played a central role – as in Andorra on Saturday – but this time joined Mason Mount alongside Declan Rice.
Mount represents a similarly adventurous option, and Southgate had recognised a need to look at "the balance of the team" after Hungary posed England "a tactical problem".
"We've got to go away and reflect," the manager said. "We shouldn't just judge things on one game in terms of that experiment, if you like, because I think right across the board, from the start, we weren't sharp with our play, gave the ball away, we were overrunning things.
"I just think we were underneath it. It's the first time in a long time, but we've got to hold our hands up to that."
Despite the difficult outing, which was further marred by clashes between police and the Hungary fans, one of whom was arrested for a "racially aggravated public order offence", England remain in control of Group I.
And discounting penalty shoot-outs, Southgate's side are now unbeaten in 18 – their longest such sequence since a run of 19 that included their 1966 World Cup win.