Murray last won the pre-Wimbledon warm-up event in 2011 and the US Open champion is eyeing another morale-boosting triumph as he prepares for his latest assault on the All England Club.
The 26-year-old is playing in his first tournament since mid-May when he suffered a lower back injury that forced him to withdraw from the French Open.
Murray had no signs of rust when he swept aside both Nicolas Mahut and Marinko Matosevic on Thursday and this was another encouraging outing against the gritty Becker.
Next up for Murray is the winner of Friday's last quarter-final between French fourth seed Jo-Wilfried Tsonga and America's Denis Kudla.
"I thought the first set was good and then he started playing much better in the second," Murray told the BBC.
"I just managed to hang tough and get the win. The conditions were tough again, but it's only my first week on grass."
The top seed had never faced Becker in competitive action before, but still knew plenty about the 31-year-old after practising regularly with him during his winter training camp in Miami.
Murray's superiority was clear as he broke in the opening game and a sloppy smash from Becker on break point in the fifth game gifted the Scot as 4-1 lead.
A brief lapse from Murray handed a break back to Becker in the next game, but the Olympic gold medallist held serve to close out the set.
Murray looked certain for a routine win when he broke at the start of the second set.
But his concentration wavered and Becker, who suddenly started to unload some searing groundstrokes, hit back with two breaks to take a 4-2 lead.
That sparked Murray back into action and he broke to force a tie-break, which he won in no-nonsense fashion.
Lleyton Hewitt rolled back the years as the Australian moved into the last four with a surprise 6-2, 2-6, 6-2 victory over Argentine third seed Juan Martin del Potro.
Hewitt is a four-time Queen's champion and this year he is bidding to become the oldest man to win the event, surpassing American legend Jimmy Connors, who lifted the trophy aged 30 years and 284 days in 1983.
He will fancy his chances after seeing off world number eight del Potro to claim his first win over a top 10 player since he defeated Juan Monaco in Valencia last October.
"I'm still hanging in there. The last four or five years have been tough with surgeries, but mentally I feel fresh," Hewitt said.
"I'm enjoying competing with the best players in the world. I played really well, I've got better with each match this week."
The former world number one, who made his first appearance at Queen's as a 17-year-old in 1998, is well into the twilight of his career and his lowly spot at 82 in the world rankings reflects his diminished status.
But the 32-year-old former Wimbledon champion enjoyed notable victories over highly-regarded Grigor Dimitrov and former Queen's champion Sam Querrey en route to the last eight and his success against del Potro earned a first last four appearance since his run to the final in Newport, also on grass, last July.
Hewitt celebrated the win by inviting his son Cruz onto the court with him.
His semi-final opponent is Marin Cilic, who hopes to retain the Queen's title after the defending champion defeated Czech second seed Tomas Berdych 7-5, 7-6 (7/4).
The Croatian fifth seed is now just two wins away from becoming the first player to retain the trophy since Andy Roddick in 2005 after a dominant display against world number six Berdych.
Results from the fifth day of the ATP tournament at Queen's Club on Friday (x denotes seeding):
Marin Cilic (CRO x5) bt Tomas Berdych (CZE x2) 7-5, 7-6 (7/4)
Lleyton Hewitt (AUS) bt Juan Martin del Potro (ARG x3) 6-2, 2-6, 6-2
Andy Murray (GBR x1) bt Benjamin Becker (GER) 6-4, 7-6 (7/3)
Jo-Wilfried Tsonga (FRA x4) bt Denis Kudla (USA) 6-3, 6-2