Wilkins was announced as Jordan boss in September, but is yet to taste victory in his three matches at the helm following a 1-0 defeat and a 1-1 draw - both against Kuwait - last month,
The home side were made to rue a glaring miss from Ahmad Hayel early in the first half with the game goalless, as Han's header shortly after the half-hour mark proved the difference between the two sides.
It is now six matches without a win for Jordan, who last tasted success in March, while for Uli Stielike's South Korea, it was a positive step after a 3-1 defeat to Costa Rica last time out.
The hosts began brightly and ought to have been in front in the 11th minute when Hayel headed Khalil Baniateyah's cross against the post from six yards with the goal gaping.
South Korea's first effort of note did not arrive until the 28th minute as Jordan goalkeeper Mutaz Yasin got down to his right to tip Park Chu-young's deflected drive around the post.
The visitors made the breakthrough six minutes later, Han stooping to head home Cha Du-ri's pinpoint delivery from the right.
Park almost doubled the advantage four minutes into the second period, flashing a shot across goal from 20 yards which went narrowly wide.
Fifteen minutes from time, Hayel had a chance to make amends for his earlier miss when he broke clear on goal, but his low shot was tipped into the side netting by Jung Sung-ryong.
That was as close as Jordan came to finding an equaliser, and Wilkins must now pick his men up in time for their friendly in Estonia on Tuesday.
