The Argentine striker was played through by Kaka to score after 21 minutes but despite their dominance Real struggled to add to their lead and they were whistled off by the Bernabeu at the end.
The biggest cheer of the night came when Cristiano Ronaldo replaced Raul midway through the second half, returning to action after two months out with an ankle injury.
Zurich did not manage to seriously trouble Real goalkeeper Iker Casillas.
The win put Real on 10 points, two ahead of AC Milan who drew with Olympique Marseille, with one round of group matches left.
"I am concerned with the quality of our play. I don't like this type of football and it's not style of play I want from the team," Real coach Manuel Pellegrini said.
"We just lacked the second goal which would have settled us and allowed us to open up. With only a one goal advantage the game was tightly balanced. The important thing was the win today."
Real camped on the edge of the Zurich area and it was Kaka who prised open their massed ranks.
The Brazilian, who had already had a long-range effort saved, danced past two players before slipping the ball into the area for Higuain and the Argentine beat Johnny Leoni with a low shot.
SUPERB VOLLEY
Xabi Alonso stung Leoni's hands with fierce drive and Marcelo cracked a superb volley against the post at a corner, but there was little else to cheer a chilly stadium.
The appearance of Ronaldo to warm up just after the break brought the home fans to life briefly, and he was roared on to to the pitch after 67 minutes.
But the Portugal forward failed to lift the team.
In the closing stages substitute Rafael van der Vaart was played through on goal but blasted over the bar with only the keeper to beat, and Leoni saved one-handed from Ronaldo after a surge into the area.
Zurich coach Bernard Challandes said: "Bearing in mind the last few weeks where we have not been doing so well in the league at home we are relatively pleased.
"We were up against the richest team in the world and we weren't humiliated. Our confidence wasn't high and it's an acceptable result for us."
