A Kerber return goes wid and Azarenka holds up her arms in triumph to win the match and go into the final; her first in three years juggling a hard tour and personal life after the birth of her son Leo.
The crowd responded with a chant of "Vika!"
Though both players served four aces and converted on 50 percent of their break point chances, Muguruza was much more effective behind her serve, as she landed almost 80 percent of her first deliveries for the match.
The Slovak was hobbled by an apparent left leg injury in the match, receiving a medical timeout after the first seven games, and her varied, counter-attacking tennis was largely diffused by the Spaniard's power game.
"I'm very pleased with how I played today", she said.
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The Belarusian had entered the final full of hope after surprising top seed Angelique Kerber 6-4 4-6 6-1 in Saturday's semi-finals.
Keys closed out the first set with a forehand service return, then broke Puig, the Olympic champion, three times in the decider.
"It's been a long time since I made this long run here", said the Dane who won the SC tournament in 2011. A backhand down the line enabled Keys to finally claim the set.
Keys won the next point but twice had backhand errors to allow Wozniacki to pull within 6-5. "I had just gotten broken and I got to go inside a reset".
The former French Open and Wimbledon champion broke the pained Belarusian three times in the first set and again in the fourth game of the second.
"It's definitely been a really great week for me, especially after having some really tough losses the last couple of weeks", Keys said.