Ranked 140th in the world, Eala, the first woman from her country to reach the last eight of a WTA 1000 tournament, took advantage of a unusually poor display from Swiatek, keeping her focus as she fought back from 4-2 down in the second set.
Eala broke the Pole for the eighth and final time when Swiatek was serving to stay in the match at 6-5 down in the second set and looked in disbelief as she struggled to contain her emotions.
"It's just surreal," said Eala, who has now beaten three grand slam winners on her route to the last four at Miami -- having dealt with Jelena Ostapenko and Madison Keys before upsetting four-time French Open champion Swiatek.
"I'm trusting my shots and I have a great team to tell me that I can do it," she said.
Eala moved to Spain at the age of 13 to join Rafael Nadal's academy in Mallorca, and Toni Nadal, the Spaniard's uncle and former coach, was in her box for the match.
"It meant a lot that he showed up here. It showed the confidence he had in me and the confidence the academy had in me," she said.
Eala will face the winner of Wednesday's quarter-final between Britain's Emma Raducanu and American Jessica Pegula.
"Just because I won this match or the one before doesn't make the next one any less tough. If anything it will be more tough, so it will take everything that I have," said Eala.
Swiatek will look forward to moving on to the clay season after conceding that she had been far from her usual level.
"I'm sure I wasn't playing my best game and I felt like my forehand collapsed a little, so it wasn't comfortable and Alexandra, for sure, used her chances and pushed me, so for sure she deserves to win this match," she said.
"I don't want to think about this too long. It's nice to learn from losses but there are other things ahead and I am happy we are going to play on clay."