After a 55-minute delay following a fierce thunderstorm five minutes into the game, both teams produced a pulsating first half on a slippery surface but second-half goals from Jeremy Menez and Yohan Cabaye after the break underlined the difference in quality.
"The conditions were the same for both teams, maybe the French team got used to the wet quicker," said Ukraine coach Oleg Blokhin.
"But the French are one of the best teams in world football and one of the favourites in this tournament," he added.
The win took France to four points, one ahead of the co-hosts who failed to find the form that brought a 2-1 win over Sweden.
The French had not won a tournament game since the semi-final of the 2006 World Cup but this was an impressive display, against a determined Ukraine backed by a loud and partisan home crowd which showed the quality now available to Blanc.
After their poor showings in Euro 2008 and at the 2010 World Cup, France are unbeaten in 23 games and it will take a very good team to stop them.
"We were on top for most of the game and I think we deserved our victory," said Blanc.
"We scored two goals, we could have scored a few more. We're improving. Can we go far [in the tournament]? I can't say. Every game is different," he said.
The artful Karim Benzema created both goals, Franck Ribery delivered direct running and pace down the left and Menez fully justified Blanc's decision to include him in place of Florent Malouda with livewire movement.
For a while though it looked as though the game might not even take place due to the sudden downpour in the opening minutes which left the field covered in large puddles and the crowd scampering for shelter.
Dutch referee Bjorn Kuipers took the players off the field in the fifth minute after thunder, lightning and torrential rain descended on the Donbass Arena and it was almost an hour before play resumed, with the surface having dried out remarkably quickly.
There was no sign, though, of the delay and loss of pre-match rhythm effecting the players as from the resumption they went for each other like two dogs who had been kept apart on a leash.
Menez had an effort ruled out for a clear offside and the pacy winger brought a brilliant save from Andriy Pyatov with a close-range strike from a low Ribery cross.
Ukraine captain and striker Andriy Shevchenko had been the hero of the win over Sweden and he showed his threat again bringing Hugo Lloris into action when he raced on to a long ball and fired a shot in at the near post.
Pyatov made another brilliant save in the 39th minute to tip over a powerful header from Philippe Mexes at the close of an absorbing and frantic first half.
After Menez brought the best out of Pyatov again, Shevchenko sounded another warning with a 49th-minute shot that curled just inches over the top of the post.
But then came three minutes of intense quality from France which settled the outcome.
Ribery burst down the left and fed Benzema who picked out Menez to his right and the winger cut inside and finally beat Pyatov with a low left-foot strike.
Benzema then threaded a sublime pass through the Ukraine defence, finding Cabaye's perfectly timed run and the midfielder held off his marker and slotted home his first international goal.
Ukraine faded in the final stages, running out of ideas and the fans showed their disappointment at a lack of effort in the final minutes, unleashing a barrage of whistles when a long ball forward was not chased down.
Blokhin's team have never won in Donetsk and the coach was unhappy with the fans after their sixth failure in the Eastern Ukrainian city.
"It is not fair to only support your team whe
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