The Croatian was imperious on serve, fending off all four break points he faced, meaning a solitary break in each set was enough for victory.
The win means Cilic has now won eight consecutive matches on the ATP Tour.
In the evening session's other match, local favourite Igor Sijsling became the first Dutchman to reach the semi-finals of the event since 2003 when he beat Germany's Philipp Kohlschreiber 6-2 2-6 6-1.
Earlier, Ernests Gulbis upset top seed Juan Martin del Potro 6-3 6-4 to move into the last four.
The unseeded Latvian was in stunning form as he outplayed the defending champion to set up a semi-final date with Tomas Berdych.
Third seed Berdych was the first man to book his place in the final four with a battling 6-7(9) 6-2 6-4 win over Polish prospect Jerzy Janowicz.
World Indoor Tournament results
Igor Sijsling (Netherlands) beat Philipp Kohlschreiber (Germany) 6-2 2-6 6-1
Marin Cilic (Croatia) beat 2-Andy Murray (Britain) 6-3 6-4
Ernests Gulbis (Latvia) beat 1-Juan Martin Del Potro (Argentina) 6-3 6-4
3-Tomas Berdych (Czech Republic) beat Jerzy Janowicz (Poland) 6-7(9) 6-2 6-4
Buenos Aires Open results
4-Nicolas Almagro (Spain) beat 8-Jeremy Chardy (France) 7-6(7) 6-3
1-David Ferrer (Spain) beat Albert Ramos (Spain) 6-1 6-2
2-Fabio Fognini (Italy) beat Pablo Andujar (Spain) 6-4 6-3
3-Tommy Robredo (Spain) beat 6-Robin Haase (Netherlands) 6-1 2-0 (Haase retired)
Memphis International results
Ivo Karlovic (Croatia) beat Jack Sock (U.S.) 7-6(4) 6-7(3) 6-3
4-Lu Yen-Hsun (Taiwan) beat Alex Kuznetsov (U.S.) 6-1 6-4
Michael Russell (U.S.) beat 3-Lleyton Hewitt (Australia) 6-3 7-6(6)
1-Kei Nishikori (Japan) beat Alex Bogomolov Jr (Russia) 3-6 6-3 6-2
- Sports & Recreation
- Tomas Berdych
- Igor Sijsling
- Philipp Kohlschreiber
- Ernests Gulbis
- Juan Martin del Potro