Kyle Walker's backheader to Brad Friedel was nowhere near strong enough and fell into the path of Beckford, whose first-time shot was pushed behind by the American keeper.
The momentum was now with the Blues and it was little surprise when they took the lead in the 65th minute with Thompson's glancing backheader looping to the net after a McAreavey free-kick.
With two minutes to go, Mohamed Fawzi was sent off for a two-footed lunge -- and Costa Rica took full advantage soon after as Urena pounced when keeper Yousif Abdulrahman could only block Allen Guevara's shot following a disastrous backheader by defender Al Kamali.