The Omani pair won 6-4, 6-7, 7-5 in a match that lasted for two hours and 41 minutes to seal the Pool C tie 2-1.
Earlier, Malaysia went up 1-0 in the first rubber when Theiviya raced past Sarah 6-1, 6-2 in just 58 minutes but Oman's No 1 player, Fatma drew parity with her 6-2, 7-5 win over Suhana Sofia.
Oman is placed along with Iran and Pacific Oceania in the Pool C in the tournament that features a total of 14 countries.
In another Pool C match, Pacific Oceania outplayed Iran 3-0.
Oman now takes on Pacific Oceania on Wednesday before completing its pool engagement against Iran on Thursday.
The winners of each pool will play-off against one another to determine which two nations will advance to Asia/Oceania Zone Group I in 2019.