Both sides enjoyed periods of dominance in energy-sapping heat in Donetsk, with England taking the lead through Joleon Lescott before Samir Nasri equalised.
“This gives us a good platform,” Hodgson said after the Group D opener.
“Hopefully we’ll get fitter. The more we train and get matches under our belts the fitter we’ll become.”
He continued: “I have to be proud of their display. No player let us down. I was confident the players would not let us down.”
Hodgson, who took charge of just his third England match and first competitive fixture, said he was pleased at how his side coped against a side unbeaten in two years. Hodgson backs England to improve
“France are one of the favourites in the group,” added the 64-year-old former West Brom manager.
“They’d gone 21 games unbeaten before today and people were tipping them to be potential winners of the tournament.
“We can’t deny on paper they might be the hardest game, but I’ve got a lot of respect for Sweden and Ukraine.
“I’ve got to be very proud of that performance. Maybe that final ball needs to improve and we were maybe a bit anxious in the final third.
“But there were good signs of a team wanting to do well.”
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“The delivery was of such good quality that it made it easy to hit the target.”
Hodgson also paid tribute to 18-year-old Arsenal winger Alex Oxlade-Chamberlain, who was the surprise selection in the starting XI.
“He did quite well, he was dangerous,” added the England coach.
“He has to be happy with his performance, but it was a great team performance. Everyone did their best to keep us in the game.
“Two or three times he skipped past players.
England face Sweden on Friday, before concluding their group fixtures against co-hosts Ukraine next Tuesday.