Mateo Kovacic calmly slotted home his effort to put Madrid on the brink of victory before left back Marcelo sealed the contest by burying his penalty into the left corner past substitute goalie Stefan Frei in front of 61,000 spectators.
Real Madrid had lost to Manchester United, Manchester City and Barcelona in their first three games of the tour and without FIFA player of the year Cristiano Ronaldo, found it difficult to find a way past starting goalkeeper Tim Howard.
The visitors finally broke through on the hour mark when 20-year-old prospect Borja Mayoral's deflected blast nestled into the right corner but Dwyer pounced on a rebound to equalise with three minutes remaining, sending the contest into a shootout.
"It was important to go back with the sensation of having won, but of course now we're going to look forward to the UEFA Super Cup which is an important game coming up," Mayoral said after being awarded the game's Most Valuable Player trophy.
Real Madrid will face Europa League winners Manchester United in the UEFA Super Cup on Aug 8 in Macedonia.
(Reporting by Larry Fine in Safety Harbor, Florida; Editing by John O'Brien)
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