Mum-of-two Maria dazzled in a 6-3 7-5 victory over the fifth seed, reaching round four of a grand slam for the first time in her career. She had suffered eight successive first-round losses in slams until this week's turnaround, last getting a win at the 2018 US Open.
The inspired Maria revelled on the Court Two big stage, becoming the oldest German woman in the Open Era, which began in 1968, to make it through to the last-16 stage of a major.
At the age of 34 years and 228 days, world number 103 Maria is also the oldest woman remaining in the Wimbledon singles this year.
Sakkari lost in two tight sets to Maria at the Australian Open in January, but the roles were reversed this time, the impressive German landing a fifth career win over a top-10 player.
Husband and coach Charles-Edouard Maria watched from the stands, while their daughters, eight-year-old Charlotte and one-year-old Cecilia, played in a nearby creche.
Maria said in an on-court interview: "I came here with my two kids for the first time. I was so happy to be in the main draw. I love to play on grass. It's such a special place for me. One year, when I was in my first pregnancy, I played pregnant.
"I love to play Wimbledon and to make this happen with my family there are no words for this, only joy and only happiness."
She said her daughters were probably "jumping in front of the TV", and spoke of wanting to set a positive example for her tennis-playing elder daughter.
"I try to be really a role model for my daughter, because she will be the next champion, so I try to show her the best way possible," Maria added. "I cannot wait to go over to the creche now and see my kids."
Neither Maria nor Sakkari had ever gone beyond round three at Wimbledon, with this the one grand slam where the Greek world number five has not reached the last-16 stage.
Maria was one of three mothers in the draw, the WTA said, along with Serena Williams and Yanina Wickmayer, both of whom made early exits. The next test for Maria will be posed by big-hitting Latvian Jelena Ostapenko.