Almagro advanced to the last four with a 6-2 6-4 victory over American Jack Sock at the clay-court tournament in Houston.
The world number 20 had endured a dry spell in his past three tournaments, with just one match win to show for almost two months of tennis.
But Almagro was ruthless early against his less-experienced opponent Sock, dropping just two points on serve as he raced to the first set at the ATP 250 tournament.
Almagro continued his serving dominance, although Sock remain on serve for nine games.
The third seed dropped just three points on his serve as he earned a 5-4 lead, and Sock could not survive a nervy 10th game - double-faulting on match point to concede defeat.
Almagro will face another American Sam Querrey in the final four, after the latter outlasted Germany's Dustin Brown 6-3 6-7 (8-10) 6-3.
Querrey got the all-important break of serve in the eighth game of the deciding set, when Brown put a backhand wide after previously saving a break point with an ace.
And the local hope ensured the Houston fans have someone to cheer for in the semis.
Almagro will be confident of progressing, as he holds a 5-2 record against Querrey - including going 2-0 against him on clay - and beat him recently at the Sony Open in Miami in March.
Meanwhile, the other semi-final will pit fourth seed Verdasco against Colombia's Santiago Giraldo in their first career meeting.
Verdasco saw off American lefty Donald Young 7-6 (7-1) 6-1 in one hour and 23 minutes.
Giraldo accounted for countryman Alejandro Gonzalez 6-3 6-4, to make a semi-final for the second time in 2014 after his run at Vina del Mar in February.