Froome finished alongside his general classification rivals, two minutes 56 seconds behind De la Cruz.
The Spaniard is not expected to challenge in the overall standings but won the 164.5km stage by 27 seconds ahead of Belgian Dries Devenyns.
He overtakes Colombian Nairo Quintana, who took the red jersey on Saturday.
Froome is aiming to become the first man in 38 years to win the Vuelta and Tour de France in the same season.
He finished alongside Quintana and second-placed Alejandro Valverde, both of Movistar, as well as three-time champion Alberto Contador after they did not challenge De la Cruz for the stage win.
1.  David de la Cruz (Spa/Etixx-Quick-Step) 3hrs 47mins 56secs
2.  Dries Devenyns (Bel/IAM Cycling) +27secs
3.  Moreno Moser (Ita/Cannondale) +33secs
4.  Luis Leon Sanchez (Spa/Astana) +51secs
5.  Mathias Frank (Swi/IAM Cycling) same time
6.  Alexandre Geniez (Fra/FDJ) +53secs
7.  Bartosz Huzarski (Pol/BORA) +58secs
8.  Thomas De Gendt (Bel/Lotto) +1min 04secs
9.  Pello Bilbao (Spa/Caja Rural) same time
10. Dylan Teuns (Bel/BMC Racing) +1min 10secs
1. David de la Cruz (Spa/Etix-Quick Step) 33hrs 46mins 24secs
2. Nairo Quintana (Col/Movistar) +22secs
3. Alejandro Valverde (Spa/Movistar) +41secs
4. Chris Froome (GB/Team Sky) +49secs
5. Esteban Chaves (Col/Orica Bike Exchange) +1min 19secs
6. Leopold Konig (Cze/Team Sky) +1min 38secs
7. Alberto Contador (Spa/Tinkoff) +2mins 01sec
8. Darwin Atapuma (Col/BMC) +2mins 06secs
9. Gianluca Brambilla (Ita/Etixx-Quick Step) +2mins 07secs
10. Samuel Sanchez (Spa/BMC) +2mins 08secs
Britain's Chris Froome slipped to fourth in the Vuelta a Espana standings as Spain's David de la Cruz won stage nine and took the leader's red jersey.