Murray raced to a first-set win in just over half an hour before Almagro suffered a recurrence of a shoulder injury and crumbled in the second to leave Murray with a superb 6-4 6-1 win.
'It was a good performance today. I served well and that helped,' said Murray afterwards.
'It was good to get it finished early. I didn't want a long match because I am playing in the Mixed Doubles this afternoon.
'Now I hope I can get a medal.'
Murray will now face Novak Djokovic, who brushed aside Jo-Wilfried Tsonga to set up a semi-final clash.
Djokovic, who has never lost to Murray in a grand slam, looked to have the trickiest of the four quarter-finals at Wimbledon, but he made light work of Tsonga this afternoon, defeating the Frenchman 6-1 7-5 on Centre Court.
Meanwhile, Roger Federer overcame the mighty serving power of John Isner to keep alive his hopes of winning Olympic gold.
The only accolade missing from the Swiss' vast collection is an Olympic Games gold in the Singles and he needed all his strength, and a large portion of luck, to beat the tall USA athlete 6-4 7-6 (7/5) on Centre Court.
Federer, who was knocked out of the men's Doubles yesterday, found it hard to deal with Isner's serves in the first set, but he broke the American in the 10th game.
A tight second set followed, resulting in a tie-break, which Federer edged thanks to a very lucky shot.
The Wimbledon champion powered down an ace to move 6-5 ahead and on the following point his return from an Isner serve clipped the top of the net and dropped over to land on the USA athlete's side while he was stranded on the baseline.
The 17-time grand slam champion apologised for the fortuitous shot before saluting the crowd, who had stood to applaud the Swiss' victory.
Federer will now face Juan Martin del Potro in the last four after the Argentinian beat Japan's Kei Nishikori 6-4 7-6 (7/4) on Court One.