"It's getting there ... with every match you start building up for the top battles starting now," said Azarenka, who only needed 57 minutes to beat the 47th-ranked Vesnina. "It's really exciting to be again so far in the tournament. "
Azarenka started last season with a 26-match winning streak, including a lopsided victory over Maria Sharapova in the Australian Open final.
She moved to No. 1 in the rankings with that win and has remained there for all but a few weeks ever since.
But she needs to defend her Australian title to hold the top spot from French Open champion Sharapova or Serena Williams, who won Wimbledon, the London Olympics and U.S. Open titles last year.
Next up for Azarenka is a quarterfinal against Svetlana Kuznetsova, who entered the season's first major ranked No. 75 but has won titles at the 2004 U.S. Open and 2009 French Open.
Serena Williams was playing later Monday against No. 14 Maria Kirilenko for a place in the quarterfinals against 19-year-old American Sloane Stephens.
Stephens advanced to the last eight of a major for the first time with a 6-1, 3-6, 7-5 win over Serbia's Bojana Jovanovski, and was looking forward to a potential match against 15-time major winner Williams.
After five breaks of serve in the third set, Stephens broke Jovanovski and then held to advance — knowing her parents and grandparents were watching on TV.
"I'm sure my mother's had like four heart attacks," Stephens joked after the see-sawing match. "I hope my Grandpa didn't have to put my Grandma to bed, because she gets a little worried."
Williams has rated Stephens as a potential world No. 1, although the young American is still yet to break through in a major.
Kuznetsova was given a time warning for taking too long during a changeover in her 6-2, 2-6, 7-5 win earlier Monday over former No. 1-ranked Caroline Wozniacki, but said it didn't bother her.
As for Wozniacki, after twice failing to win her breakthrough major as the No. 1-ranked woman at the Australian Open, she is now expected to drop out of the top 10 after a fourth-round loss that follows back-to-back first-round exits at Wimbledon and the U.S. Open.
Kuznetsova is moving in the other direction on her comeback from a right knee injury that ruled her out of the last U.S. Open, ending a run of contesting 40 consecutive majors. And she's looking forward to taking on Azarenka.
"The time I played in Indian Wells was totally a disaster. I got (beaten) very badly," she said. "But, I mean, she's tough; she's No. 1; I have nothing to lose; she has all the pressure.
"I know I got the game to give her some problems, and I will just do my best and just try to enjoy it."
On the men's side, Frenchman Jeremy Chardy followed up his upset win over 2009 U.S. Open champion Juan Martin del Potro by beating another higher ranked player in the fourth round, defeating No. 21 Andreas Seppi 5-7, 6-3, 6-2, 6-2 on Monday.
He'll next play the winner of Monday's later match between No. 3 Andy Murray and No. 14 Gilles Simon of France.
Another Frenchman already through was seventh-seeded Jo-Wilfried Tsonga, who beat his old friend Richard Gasquet 6-4, 3-6, 6-3, 6-2 in the fourth round.
Tsonga, who lost the 2008 Australian Open final to Novak Djokovic, could meet 17-time major winner Roger Federer in the quarterfinals.
Play on Day 7 ran into Day 8 overnight, when Novak Djokovic needed just over 5 hours to beat Stanislas Wawrinka 1-6, 7-5, 6-4, 6-7 (5), 12-10 in a fourth-round match Sunday night, on the same court where he needed 5:53 to beat Rafael Nadal in last year's final.