The American top seed won 6-0 6-3 at the Hisense Arena but needed eight set points to clinch the opener, while Suarez managed a comeback of sorts in the second set, winning three games.
However, there was clearly only going to be one winner and Williams moves on to face Australian Sam Stosur in the last 16.
Williams seemed on course for a first-set whitewash as she strolled to a 5-0 lead, with Suarez winning only five points in that time.
The Spaniard came to life then, however, delaying the inevitable in a 23-minute sixth game which saw 13 deuces and seven wasted break points by the American.
Boosted perhaps by that resistance, Suarez's intelligent shot selection saw her push Williams harder in the second set.
Trademark
The 11-time Grand Slam champion displayed her trademark power and stamina, however, taking everything Suarez could muster and, after breaking her opponent in the third and reaching 5-3, broke the Spaniard again to wrap up the win.
The 28-year-old Californian is expecting a tough challenge in the next round where she will meet Stosur, a 6-4 6-1 winner over Alberta Brianti.
Stosur beat Williams when they last met in Los Angeles last year and with a noisy home crowd behind the Australian, the defending champion is wary of the threat to her crown.
"She beat me the last time we played, so she has a lot of momentum," said Serena Williams.
"A ton of momentum. She serves great. She gets a lot of big, big first serves. I definitely expect the crowd to root for Sam. I mean, I would."
Stosur claimed she could cause the upset, saying: "I guess I don't really have that much to lose. I can go out there and just really enjoy it."
Hat-trick
Venus Williams also progressed as she made it a hat-trick of wins against left-handers by overcoming Australia's Casey Dellacqua.
Williams won 6-1 7-6 but was made to battle in a contest which lasted one hour and 47 minutes.
It was the sixth seed's third win against left-handed opposition, after beating Lucie Safarova and Sybille Bammer in the earlier rounds, and - like her sister -she is still to drop a set in Melbourne.
The final match on Saturday saw No.4 seed Caroline Wozniacki advance to the fourth round after blitzing Israel's Shahar Peer 6-4 6-0.
The talented Dane, a beaten finalist at the US Open four months ago, will play Na Li for a place in the last eight after the Chinese player overcame Slovakia's Daniela Hantuchova 7-5 3-6 6-2.
