Top seed Murray, bidding for a record-equalling fourth title at the prestigious Queen's Club, fended off some typically ferocious hitting by Spaniard Verdasco to win 7-5 6-4 and delight a large Centre Court crowd.
Earlier defending champion Grigor Dimitrov, who stunned Briton Murray in the Wimbledon quarter-finals last year, became the latest high-profile seed to fall, losing 6-4 7-6(5) to Muller.
Bulgarian Dimitrov, the sixth seed, struggled to deal with Muller's dangerous serve as he joined French Open champion Stanislas Wawrinka and 14-times major winner Rafa Nadal on the casualty list of a tournament boasting increased prize money and ranking points this year.
The 32-year-old Muller has served 56 aces in two matches so far and will now take aim at Murray for a place in the semis.
"It's these challenges that I'm working for to play those matches on the big courts against the best guys in the world," Muller, who has battled his way back into the top 50 after sliding to 374 in the ATP rankings in 2013 after elbow problems.
Muller broke in the 10th game when Dimitrov made some sloppy mistakes and then hit back from 0-3 in the second set tiebreak to clinch victory with a wrong-footing forehand winner.
Murray, dressed head to toe in white, made a fast start against left-hander Verdasco and marched into a 3-0 lead.
Verdasco, who was two sets up against Murray in the Wimbledon quarter-finals two years ago before the Scot came back to win and went on to claim an historic title, warmed to his task though and levelled it up at 4-4.
Murray took the opening set when Verdasco netted a forehand in the 12th game but had to fend off several break points early in the second set, coming back from 0-40 down at 1-1.
Serving at 4-5 Verdasco self-destructed, serving three double faults before seeing two aces chalked off on match point down because the ball brushed the net tape whizzing through.
Another forehand error sealed his fate as Murray took his career record over the 31-year-old to 11-1.
"It was a tricky match. Fernando is obviously an extremely powerful guy and you don't have a chance to get into some of the rallies," world number three Murray said on court before looking ahead to his next opponent.
"Gilles Muller served 37 aces in the first round -- I haven't served that many all year!"
(Reporting by Martyn Herman, editing by Tony Jimenez and Toby Davis)
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