The German came from a set down to win 6-7 (3-7) 7-6 (7-5) 6-4 in s-Hertogenbosch against Canadian Pospisil.
Becker will face qualifier Joao Sousa in the last four of the competition after the Portuguese world number 48 beat wildcard Thiemo de Bakker 6-4 6-2.
There was nothing to choose between world number 83 Becker and Pospisil as the first set went with serve, but the Canadian took charge of the tie-break to lead 6-1 and although Becker claimed two mini-breaks he was unable to complete an unlikely fightback.
Becker rallied in the second set as he broke to take a 2-0 lead but Pospisil broke back and although the German fashioned three set points, his opponent stood firm to ensure a second tie-break ensued.
A solitary mini-break was sufficient for Becker to level the match and one break in the decisive set sealed his progress into the last four.
Sousa was not given such a stern test en route to the semi-finals as he proved to be too strong for De Bakker.
De Bakker only had one break point in the match and was unable to take it as Sousa broke to go 3-2 up in the first set and there was no way back for his opponent, who then took control of the second set with a further two breaks and eased into the semis.
Jurgen Melzer and Fernando Verdasco were locked at a set apiece when bad light called a halt to the match after Melzer took the first set 7-3 on a tie-break and the Spaniard won the second-set tie-break 8-6.
Nicolas Mahut and Roberto Bautista Agut were level at 3-3 in the first set when play was halted due to darkness, so both games will resume on Friday.
The German came from a set down to win 6-7 (3-7) 7-6 (7-5) 6-4 in s-Hertogenbosch against Canadian Pospisil.
Becker will face qualifier Joao Sousa in the last four of the competition after the Portuguese world number 48 beat wildcard Thiemo de Bakker 6-4 6-2.
There was nothing to choose between world number 83 Becker and Pospisil as the first set went with serve, but the Canadian took charge of the tie-break to lead 6-1 and although Becker claimed two mini-breaks he was unable to complete an unlikely fightback.
Becker rallied in the second set as he broke to take a 2-0 lead but Pospisil broke back and although the German fashioned three set points, his opponent stood firm to ensure a second tie-break ensued.
A solitary mini-break was sufficient for Becker to level the match and one break in the decisive set sealed his progress into the last four.
Sousa was not given such a stern test en route to the semi-finals as he proved to be too strong for De Bakker.
De Bakker only had one break point in the match and was unable to take it as Sousa broke to go 3-2 up in the first set and there was no way back for his opponent, who then took control of the second set with a further two breaks and eased into the semis.
Jurgen Melzer and Fernando Verdasco were locked at a set apiece when bad light called a halt to the match after Melzer took the first set 7-3 on a tie-break and the Spaniard won the second-set tie-break 8-6.
Nicolas Mahut and Roberto Bautista Agut were level at 3-3 in the first set when play was halted due to darkness, so both games will resume on Friday.