The duo traded breaks early in the contest before world No.96 Griekspoor, gained a foothold to go 4-2 up and the 26-year-old held firm on serve thereafter to take the first set when Thompson sent a backhand wide.
Thompson, ranked 84, was more aggressive in a tighter second set and went 6-5 up with a service hold before breaking a tiring Griekspoor in the next game to drag himself into the contest.
Backed by his Australia team mates and the crowd at the Palacio de deportes Martin Carpena, the 28-year-old Thompson turned up the heat to establish a comfortable lead and held off a late comeback attempt by his opponent to take the victory on serve.
Alex de Minaur can now secure Australia's place in the semi-finals with victory over Botic van de Zandschulp in the second singles match.
However, defeat for the world number 24 will mean the teams will play a deciding doubles rubber where Matthew Ebden and Max Purcell face Dutch pair Wesley Koolhof and Matwe Middelkoop.
The winners of Tuesday's tie will face either six-times champions Spain or Croatia for a place in Sunday's final.
In other quarter-finals, Italy take on the United States while Germany face Canada.