The loss was the first time in over 60 years that Hungary have beaten England and they could have won by more if Andras Schafer had converted a late chance.
Reece James gave away the penalty just after the hour mark, a mere two minutes after he was substituted onto the pitch.
"The heat was a factor. That took a lot out of the players," Southgate said after the game. "I don't want to be too harsh on them. These are games we need to learn from."
Harry Kane, who was looking for his 50th international goal, had some good chances but failed to put any away.
"It's a massive disappointment," defender Conor Coady said after the game. "It's something we need to look at and learn very quickly.
"They're a good side, they're organised, you can see why they've got results, but we wanted to come here and win.
"We started off quite well but they grew into the game a bit.
"We were shocked because nobody appealed for the penalty... but we need to look at ourselves."
The fixture in Budapest was technically a "behind closed doors" match following UEFA sanctions issued after anti-gay and racist chanting by Hungary fans during their three Euro 2020 group stage matches.
However, a 35,000 crowd made up predominantly of children was allowed and fans booed England players taking the knee before kick-off.
"It's massively disappointing," Coady said. "It's important that people understand why we're trying to do it... You don't want to hear them things, but it's important we keep on doing what we're doing, to make people talk about the situation.
"It is what it is, but we want to keep pushing that message."