The four-time grand slam champion was made to work harder than Friday's scoreline suggests but was still relatively untroubled as she matched her best run at the championships.
She will face Swiss 14th seed Belinda Bencic next after the Olympic champion claimed a resounding 6-3 6-1 win over Poland's Magda Linette.
Service breaks were traded early in the first set before Swiatek reeled off four games in a row.
The 30th seed Martic kept in touch with Swiatek early in the second set but again the world No.1 managed to find some extra gears to move towards victory.
Martic had managed to win just seven games in two previous meetings with Swiatek but doubled that total after she saved a match point at 3-5 before breaking the Swiatek serve to love.
The momentum was shifting as Martic then went 40-0 ahead on serve for the chance of a 6-5 lead but in the blink of an eye Swiatek hit back to break and wrapped up victory in the next game as Martic bungled a drop shot into the net.
"I'm happy I could close it out at the end of the second set," the 22-year-old Swiatek, who has been the top women's player in the world for 67 successive weeks, said on court.
"Belinda is playing a great game and it's never going to be easy in the fourth round of a slam."
Tunisian Ons Jabeur made quick work of Chinese qualifier Bai Zhuoxuan with a 6-1 6-1 victory in 45 minutes to advance.
Jabeur is looking to go one step further at Wimbledon this year after losing the final in 2022 and the sixth seed was flawless with attacking shots that sent her opponent, who is ranked 185 rungs below her, scrambling.
Fifth-seed Caroline Garcia was upset by Czech Marie Bouzkova who won 7-6 (7-0) 4-6 7-5, her third victory over the Frenchwoman in as many games on grass.
The match was level at one set apiece when organisers decided to move the contest due to bad light from Court Two to Court One, which is lit under a roof, where Garcia lost all the momentum she had gained in the second set.
The Frenchwoman squandered a break in the decider and, although she saved a match point, Bouzkova prevailed to set up a fourth-round clash with compatriot Marketa Vondrousova.
"This match had pretty much everything, my heart rate is at 300 right now!" said Bouzkova, who is seeded 32.
"It wasn't easy. We just had 15 minutes to move here in a completely different and bigger court, I just tried to keep my head in the game and take a lot of caffeine."
Lesia Tsurenko dropped to the ground in exhaustion following her Wimbledon third-round win over Ana Bogdan, in which she saved five match points and won the longest tiebreak in women's singles grand slam history.
The Ukrainian and her fellow unseeded Romanian opponent split the opening two sets but an incredible battle for the ages unfolded at 6-6 as Tsurenko dug deep into her reserves to win 4-6 6-3 7-6 (20-18).
Tsurenko saved five match points and converted her seventh to secure victory after a lengthy final point, prompting the crowd on Court 14 to cheer wildly as the Ukrainian lay on the ground breathing heavily as her opponent waited at the net.
The pair shared a warm hug after a match lasting three hours and 40 minutes.
The previous longest tiebreak in a women's grand slam singles match was 36 points, in the opening set of Denisa Allertova's win over Johanna Konta in the first round of the 2015 French Open.
Former Wimbledon semi-finalist Elina Svitolina earned a 7-6 (7-3) 6-2 win over Sofia Kenin while fourth-seed Jessica Pegula claimed a comfortable 6-4 6-0 win over Italian Elisabetta Cocciaretto.
Second seed and Australian Open champion Aryna Sabalenka fought back from a set and 5-4 down to beat Varvara Gracheva 2-6 7-5 6-2.
Twice Wimbledon champion Petra Kvitova beat Aliaksandra Sasnovich 6-2 6-2 with a flurry of winners while Madison Keys defeated Viktorija Golubic 7-5 6-3.
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