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."
Hodgson, who took charge of his third England match and first competitive fixture, said he was pleased at how his side coped against a team unbeaten in two years.
"I have to be proud of their display. No player let us down. I was confident the players would not let us down," added the 64-year-old former West Bromwich Albion boss.
Euro 2012: England players happy with France draw
"France are one of the favourites in the group. They'd gone 21 games unbeaten 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."
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.
"I said to him before the game that it's a big occasion, but there will be several more big occasions so don't get too hung up about this. Do what you do. There will be lots more games for him in an England shirt."
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