The Welsh looked destined for defeat in their first appearance in the tournament for 64 years after Timothy Weah opened the scoring in the first half.
But their talisman Bale won and scored the spot-kick to earn them a precious point in Group B.
They next face Iran on Friday morning, while the USA take on Group B leaders England in the evening on the same day – after Gareth Southgate’s side won 6-2 earlier on Monday.
Bale is the difference again for Wales, it was the USA who dominated the early stages and could have taken the lead on nine minutes as Joe Rodon’s inadvertent header towards his own goal was kept out by Wayne Hennessey, before Josh Sargent nodded against the post shortly after
The breakthrough came on 36 minutes as Christian Pulisic found space to slip a pass in behind for Weah, who smartly jabbed the ball past Hennessey and into the bottom corner.
Wales had been overwhelmed in the first half and in response manager Rob Page brought on Kieffer Moore for the ineffective Dan James at the break
And the Bournemouth striker should have levelled for Wales on 65 minutes, as he rose at the near post from a corner, but somehow headed over with the net at his mercy.
With eight minutes to go they did find the equaliser, however. Bale got himself in front of Walker Zimmerman in the box to be tripped and win a penalty, before stepping up himself to thunder the ball past Matt Turner.
The result leaves both sides trailing England in Group B, after they earned all three points against Iran on Monday afternoon.