A highly consistent first 11 races for Hamilton have left him with hope of claiming a record eighth drivers' title, as he sits 50 points behind Mercedes' Kimi Antonelli at the top of the standings.
While the gap is far from insurmountable, Antonelli's Mercedes has generally appeared to have a performance edge of Hamilton's Ferrari, meaning the Italian squad likely need to find further gains if the 41-year-old is to maintain his challenge.
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Ferrari also remain in the hunt for the constructors' title, with Mercedes only 72 points clear despite their early-season dominance.
Vasseur said: "The first half of the season was in stages. Until Spain, Mercedes was flying.
"After Spain, when we came back, we scored more points than Mercedes in the last four or five races."
The Frenchman was speaking after the final round before the summer break in Hungary, during which Ferrari and Hamilton made a series of errors that saw him finish fifth when the car was at least fast enough to be in the podium places.
"Where do we have to improve? If I draw a conclusion based on Hungary, I would say execution," Vasseur continued.
"But execution so far was probably a good one for us from the beginning of the season."
"We know that we have to improve on the engine side and we will do it. The inertia on the engine is much higher than on the chassis."
Why Vasseur thinks Ferrari can respond to McLaren surge
It has been suggested by rivals, including McLaren's Hungarian Grand Prix Lando Norris, that Ferrari have the best chassis on the grid, but have been held back by their power unit not being as strong.
Ferrari have already once upgraded their engine having been granted the opportunity to do so by the FIA's performance-balancing measures, but it remains to be seen how much more improvement can be brought to the power unit over the second half of the season.
While Vasseur acknowledges there is room for improvement of the engine, he insists the team must continue to work hard to bring further upgrades to the chassis.
"The most important thing will be to be able to keep the pace in the development," Vasseur said.
"McLaren did a good job for a while, but they don't have exactly the same philosophy [as us ]because they are bringing a big package.
"For sure each time they bring something, they are making a huge step. We have a different approach, trying to bring parts every race.
"We have to keep this momentum because there are still 12 races to go. It's a lot. We have a room for improvement on the chassis everywhere, and we have to keep this pace."
The Formula 1 season resumes after the summer break with a Sprint weekend at the Dutch Grand Prix from August 21-23, with every session live on Sky Sports. Stream Sky Sports with NOW - no contract, cancel anytime
