Commenting on his goal, Weah said, "Christian gave me a beautiful ball and it was up to me to finish it."
A late penalty by veteran striker Gareth Bale earned Wales a 1-1 draw with the United States on Monday to avoid a losing return to the World Cup after 64 years in a fascinating Group B encounter at the Ahmad Bin Ali Stadium.
Dragged down by U.S. defender Tim Ream in the box, Wales' all-time top scorer stepped up to the penalty spot in the 82nd minute and made no mistake by hammering the ball into the top corner.
They were unlucky not to go into the break with a bigger advantage against the flat Welsh.
Weah, the son of Liberian president and former World Player of the Year George Weah, timed his run to perfection after a beautifully-weighted pass from Christian Pulisic and slipped the ball past Wales goalkeeper Wayne Hennessey.
Commenting on his goal, Weah said, "Christian gave me a beautiful ball and it was up to me to finish it."
The forward added on the result: "It felt like in the first half we had a lot of energy, a lot of momentum ... and then coming into the second-half we dialed down and Wales turned it up a notch. They started pressing us, they had most of the ball and I think at the end that’s what really hurt us."
Wales manager Rob Page hailed Gareth Bale and revealed he had been considering substituting his talisman in the second half of the 1-1 draw with the United States on Monday before his late penalty earned them a precious point in Group B.
Bale, 33, helped Los Angeles FC win the MLS Cup this season but has yet to play a full match for the Major League Soccer side and entered the World Cup having played only 28 minutes of competitive football since September.
"Irrespective of what he does at club level, he manages to put the jersey on for Wales and give performances that surprise people. And he certainly surprised me today," Page told reporters.
"I said to him on about 75 minutes, 'Are you OK to carry on?' And he went: 'Yes, I'm fine.' And then at the end of the game he said, 'That's why I said I'm fine.'