Real, starting their bid for a 10th European title with new coach Jose Mourinho at the helm, totally outplayed the Dutch four-times winners but squandered a slew of chances at their sultry Bernabeu stadium.
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Ajax hardly troubled Spanish World Cup-winning captain Iker Casillas in the home goal and disappointed coach Martin Jol described the contest as "boys against men".
"My attackers maybe weren't on their best form in front of goal but we completely dominated the match," said Mourinho, poached from Inter Milan after leading the Italians to a treble of domestic league and cup, and Champions League last season.
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"This is the group phase so the goals are not as important as knockout games and I am happy with the three points," the Portuguese added. "It gives us some breathing space."
Mourinho and his players were under pressure to entertain their demanding supporters after a couple of lacklustre displays in the Spanish league and a sparkling 5-1 win for arch rivals Barcelona against Panathinaikos on Tuesday.
However, Portuguese forward Cristiano Ronaldo and Argentina winger Angel Di Maria were especially wasteful and Ronaldo, the world's most expensive player, was the target of impatient whistling from some sections of the crowd for his profligacy.
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In contrast, Spain's new U.S. Open champion Rafa Nadal, a Real fan and a shareholder in Real Mallorca in his native Balearic Islands, enjoyed a standing ovation when he strolled on to the pitch before kick-off.
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Di Maria's international team mate Higuain was initially credited with the opening goal in the 31st minute but UEFA later ruled it had gone in off defender Vurnon Anita.
New fan favourite Ozil again impressed in the creative role vacated by the injured Kaka and the German international - bought from Werder Bremen in the close season - set up Higuain to make the game safe in the 73rd minute.
Netherlands goalkeeper Maarten Stekelenburg blocked Ozil's initial strike but his quick thinking enabled him to play the rebound across goal for an unmarked Higuain to score.
Ozil, one of the best performers at the World Cup in South Africa, was given a rousing ovation when he was replaced late in the game and earned warm praise from his coach.
"The kid is really good and the important thing is the personality he has," Mourinho said.
"To play for Real is not easy and to be loved by the Bernabeu is not easy and it looks like the kid will play for Real for many years."
The closest Ajax came to scoring was when Urby Emanuelson hit the crossbar with a speculative effort with just over 10 minutes remaining.
Real had 36 shots overall and 16 on target while Ajax mustered only one decent effort on goal.
Former World Player of the Year Ronaldo capped a frustrating night when he somehow let the ball slip between his legs with the goal at his mercy in the 89th minute.
Higuain acknowledged that Real were less than sparkling: "We can play better. We need a bit more luck with the chances we create," he told local television after the match.
"We carved a lot of openings and the scoreline could have been much higher in our favour. But we are pleased. It's still early in the season and we have a lot to put right but we have started well."
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