Kevin Diks had put Indonesia ahead from the spot in the 11th minute but Saleh Abu Al-Shamat levelled six minutes later before Al-Brikan scored either side of halftime to secure the points despite a second Diks penalty two minutes from time.
Saudi Arabia next face Iraq in Jeddah on Tuesday and know that a win over Graham Arnold's side will guarantee their place in December's draw for the finals in the United States, Canada and Mexico.
Indonesia and Iraq will face one another on Saturday.
The Indonesians went in front when Hassan Al-Tambakhti stuck out an arm as Dean James swung in his free kick from the right and Diks stepped up to clinically place his spot-kick beyond Nawaf Al-Aqidi.
The lead was to last only six minutes, however, as Al-Shamat steered his right-foot strike from the edge of the area beyond Maarten Paes' dive to his left.
With nine minutes left in the half, the home side went in front. Yakob Sayuri's needless pull on Al-Brikan's shirt saw referee Ahmad Al-Ali called to the pitch-side monitor and the Kuwaiti awarded a penalty that Al-Brikan slammed home.
Al-Brikan put the result beyond doubt in the 62nd minute with a clinical close-range finish.
Paes threw himself to his right to keep out Musab Al-Juwayr's low drive only for Al-Brikan to pounce on the rebound and send the ball back under the body of the Indonesia goalkeeper.
Diks struck again from the penalty spot after Nawaf Bu Washl hit the ball with his arm as he went down in his own area in a tussle with Ole Romeny, while Mohammed Kanno's injury-time red card made for a nervous finish for Herve Renard's side.
Six of Asia's eight guaranteed World Cup berths have already been filled by Japan, South Korea, Australia, Iran, Jordan and Uzbekistan, with a possible ninth spot available at the 48-team finals via an intercontinental playoff in March.