Playing in temperatures approaching triple digits with high humidity, Raymond had a quick start and a strong finish to turn back the former Wimbledon champion in just over 2 1/2 hours. She overcame a 3-1 deficit in the third set.
"It was a tough match and it was battle of wills in the end," Raymond said. "The clouds came over in the third, but the first couple sets were brutal (in the heat). You just had to work for every point out there. I just capitalized on coming in and playing the big points a little better than she did today."
After winning the first four games and taking the first set, Raymond won five straight games in the second set to take a 6-5 lead. But she squandered a pair of match points and lost the tiebreaker.
Following a 10-minute break in accordance with the WTA's heat rule, the 30-year-old Martinez carried the momentum into the third set. But Raymond broke back and tied the match, 3-3.
"It was just a matter of hanging in and getting my chances," Raymond said. "When I was down 3-1 in the third, I knew I could break."
In the semifinals, Raymond will face Els Callens of Belgium, who rallied for a 4-6, 7-5, 6-0 win over American Samantha Reeves.
The other semifinal pits Spain's Gala Leon Garcia against Zimbabwe's Cara Black, which assures an unseeded player of reaching the final.
Leon Garcia cruised to a 6-0, 6-3 win over American Jill Craybas. Black eliminated her doubles partner, the fourth-seeded Elena Likhovtseva of Russia, 6-4, 6-0 to reach her first WTA semifinal of the year.
Swim Week Miami Beach 2014 [PHOTOS]