A single break in the first set put the Polish top seed in charge and she repeated the feat in the second set to seal a 6-4, 6-3 victory.
Martic needed treatment for an apparent lower back issue midway through the first set of the second-round match on Centre Court before resuming with no apparent discomfort.
Swiatek broke her 85th-ranked opponent in the 10th game to take the first set 6-4.
Martic battled hard in the second set but Swiatek ultimately had too much power, breaking in the eighth game from 40-0 down and serving out to love for the win.
Swiatek hit 25 winners against 19 unforced errors in the match, which lasted 83 minutes.
The top seed, who has never been beyond the quarter-finals at Wimbledon, is chasing a sixth Grand Slam title following her recent French Open triumph.
She skipped the Berlin grass-court event last month to rest, meaning she came to Wimbledon with no matches on the surface under her belt.
The 23-year-old, who won the girls' title at the All England Club in 2018, will face Kazakhstan's 35th-ranked Yulia Putintseva for a place in the last 16.
