Benayoun struck in the seventh minute with a low shot from close range, converted a penalty in the 64th and finished off with a header 15 minutes from time.
Malta's Jamie Pace headed an equaliser in the 38th minute to send the teams in level at half-time but the visitors found their hosts too aggressive and skilful throughout the match.
The other opening Group F fixtures will be played on Friday when Latvia host Croatia and Greece are at home to Georgia.
Benayoun set the tone for a new-look Israel side under the guidance of French coach Luis Fernandez and he also had an effort disallowed.
Malta never looked like threatening Israel's dominance at their home bastion where they rarely lose and had to rely on a handful of counterattacking moves either side of halftime when they were at their most dangerous.
The hot and humid conditions helped neither team but Fernandez insisted that Israel had faced tough opposition.
"I told you it would not be an easy match, remember what I said... if you came to see eight goals, it was not going to be that sort of game," Fernandez told a news conference.
"There are many things that need to be improved upon, we have two years to try to qualify, these are very tough games and it was also very hot," the Frenchman added.
"It will be harder and more complicated against Georgia, they are a very talented side," Fernandez said of Israel's next opponents whom they play away on Tuesday.
Malta coach John Buttigieg lamented two major errors.
"We knew that it would be a difficult match, Israel put us under pressure ... but unfortunately we made two big mistakes towards the end. We are very disappointed but I think Israel deserved to win," he said.
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