Nader Al Enezi struck in the 47th minute to give Kuwait their second victory in three matches. Saudi also have six points but trail Kuwait on head-to-head record.
Uzbekistan lead the standings with nine points and face the Saudis in their final match on Tuesday.
Kuwait will be expected to beat winless Afghanistan in their last game. Victory for Saudi and Kuwait would leave three teams on nine points, which means goal difference would then apply.
In the other match on Sunday, the Afghans drew 1-1 with Pakistan as both teams picked up their first points of the qualifiers.
Zemarai Mohammadi gave Afghanistan a 21st-minute lead but Faheem Ahmed equalised for Pakistan with a minute left.
The top two teams and best third-placed team from Groups A-G qualify for the finals along with the winners of Group H.