Higgins looked on course for comfortable progression as he led Gould 7-2 overnight, but the world number 18 fought back and closed the gap to 8-5.
However, 41-year-old Higgins resisted the fight back to go through to the second round where he will play Northern Ireland’s Mark Allen.
"To win 10-6 against Martin is brilliant," Higgins said. "He's a class player and at 7-2 up that brings pressure. Martin can roll off two or three frames very quickly.
"Had it gone 9-7 it would have been difficult to get over the line. I am very happy to get through."
Meanwhile world number four Ding Junhui also won his first-round match when seeing off compatriot Zhou Yuelong 10-5.
"I played well in the first session when he was a bit nervous. It was hard for him because he has no experience to play at the Crucible," said Ding, who now plays another fellow countryman, Liang Wenbo, in the second round.