Rain delayed the start of Murray's match but the 18-year-old Scot needed just 47 minutes to book his second-round berth.
Murray was quick to capitalise on each of his three break points in the first set.
Mazarakis managed a break of his own in the second, but the British number one stormed home by winning 24 of the 33 points played.
Stadium 2 start 10:00 AM
G Rusedski (GBR) vs S Wawrinka (SUI)
W Moodie (RSA) vs P Srichaphan (THA)
Not before 12:00 NOON
J Bjorkman (SWE) vs C Rochus (BEL)
A Seppi (ITA) vs P Mathieu (FRA)
(Q)B Mattek (USA) vs S Mirza (IND)
Murray (GBR) vs (5)N Davydenko (RUS)
S Foretz (FRA) vs A Sugiyama (JPN)