Verdasco dumped early in Brisbane
The Spaniard, a semi-finalist at the Australian Open in 2009, recovered from a listless start to grind out the second set but had few answers in the third as Becker played a faultless service game to seal the match in just over two hours.
"I had a good game plan and it worked well today," 53rd-ranked Becker said after setting up a second round match with Colombia's Santiago Giraldo.
"It is always good to get a win under your belt... and it's not every day you beat a top 10 player."
The 27-year-old Verdasco will have one more competitive warm-up before the January 17-30 Australian Open - at the invitational Kooyong Classic in Melbourne next week, where he will defend his title.
After beating Frenchman Jo-Wilfried Tsonga in the final at Kooyong last year, the feisty Spaniard advanced to the fourth round at Melbourne Park but was edged in a five-set thriller by Russia's Nikolay Davydenko.
Top seed Robin Soderling had a more satisfactory day in Brisbane, notching a comfortable 6-2 6-4 win against American qualifier Ryan Harrison.
The world number five faces Michael Berrer, of Germany, in the next round.
Fourth-seed Mardy Fish also advanced to the second round - with a 6-1 6-4 victory over France's Adrian Mannarin - where he is joined by 2009 champion Radek Stepanek and world number 20 Marcos Baghdatis, who eased to first round victories over Germany's Tobias Kamke and Florent Serra respectively.