The Briton is trying to be the first rider in 38 years to win the Tour de France and Vuelta in the same year.
Another of Froome's key rivals, Nairo Quintana, finished four-hundredths of a second adrift with his Movistar squad.
Manxman Pete Kennaugh led the Team Sky squad over the finish line to take the race leader's red jersey.
Just five of Sky's nine-man squad crossed the line together, the minimum needed to register a time.
Spaniard Contador was the last man to win two of the three main road races in a calendar year, winning the Giro d'Italia and Vuelta in 2008.
Only Jacques Anquetil, in 1963, and Bernard Hinault, in 1978, have previously won the Tour de France and Vuelta in the same season.
"To start almost a minute from direct rivals is more than I imagined," said Contador, who is looking to win the Vuelta for a joint record fourth time.
"It's a big deficit. There are still 20 stages to go and we will try to reduce the gap day by day."
Sunday's 159km second stage takes the riders from Ourense to Baiona.
Team time trial classification:
1. Team Sky (GB) 30mins 37secs
2. Movistar (Spa) same time
3. Orica BikeExchange (Aus) +6 secs
4. BMC Racing (US) +7secs
5. Etixx - Quick-Step (Bel) +22secs
6. Trek-Segafredo (US) +50secs
7. Cannondale (US) +52secs
8. Tinkoff (Rus)  Same time
9. BORA Argon (Ger)  +57secs
10. Astana (Kaz)  +58secs
General classification:
1. Peter Kennaugh (GB/Team Sky) 30mins 37secs
2. Salvatore Puccio (Ita/Team Sky) same time
3. Michal Kwiatkowski (Pol/Team Sky)
4. Leopold Koenig (Cze/Team Sky)
5. Chris Froome (GB/Team Sky)
6. Alejandro Valverde (Spa/Movistar)
7. Jonathan Castroviejo (Spa/Movistar)
8. Ruben Fernandez (Spa/Movistar)
9. Jose Joaquin Rojas (Spa/Movistar)
10. Nairo Quintana (Col/Movistar)
Selected others:
14. Simon Yates (GB/Orica BikeExchange) +6 secs
50. Alberto Contador (Spa/Tinkoff) +52 secs
124. Scott Thwaites (GB/BORA Argon) +02mins 01sec
179. Hugh Carthy (GB/Caja Rural) +04mins 34secs
Chris Froome opened a 52-second lead over rival Alberto Contador after Team Sky won the team time trial on the opening stage of the Vuelta a Espana.