Hodgson's team emerged with a valuable point from a Group D encounter played in furnace-like conditions at the Donbass Arena after Samir Nasri's strike cancelled out Joleon Lescott's opener. England appeared to have suffered from the heat more than France, who finished the game on top but were unable to find a winner.
Hodgson believes the result will act as a springboard for a successful first round campaign however, with England facing Sweden in Kiev on Friday before taking on Ukraine in their last Group D match. "I hope this gives us a good platform. We'll get fitter as well - match fitter," Hodgson said.
Quite a few of our players didn't play the last few games at the end of the season and didn't come into this tournament with the same match fitness they might have done. "The more we train and play, the fitter they'll become. It was tough in those last 15 or 20 minutes.
"It was fortunate I had some players to put on to give them a break. But Laurent [Blanc, the France coach] will also say that both teams were tired. It wasn't one-sided. The French were also feeling the effects of the heat."
Hodgson meanwhile stressed that while his team would grow in confidence they still had plenty of room for improvement after what was only the England boss's third match in charge. "I hope we've established ourselves [as hard to beat], but I've had three games," Hodgson said. "Am I satisfied with those three games, and have the players done what's asked of them? Then yes.
"But we've got Sweden now. Who knows? Let's wait and see. You don't become a really good team in three matches and ten training sessions.
"The French have gone 22 games unbeaten and they've not done that overnight," he added. "It's been people playing together, getting to know each others' games. The longer we play together, the better we'll become too."