The Tinkoff rider finished the 4km time trial ascent of Mont Chery in the French Alps in 11 minutes, 36 seconds.
Australia's Richie Porte, riding for BMC Racing, was six seconds adrift, with Team Sky's Froome third.
Britain's Froome has won the race twice before - in 2013 and 2015 - both times going on to win the Tour de France.
He decided to go out early in the race against the clock because he was concerned it might rain later in the race.
"It was tough. I was running out of legs," he said before Contador and Porte had finished.
"I'm happy with that. Just the feeling makes me happy. I feel like I'm in  good shape. I still have some work to do and I hope to keep the momentum till the Tour de France."
Contador, also a two-time Tour winner (2007 and 2009), is yet to win the eight-stage, week-long Dauphine, but has twice finished second - in 2010 and 2014.
He said: "The climb suited me really well. It's a bit of a surprise to beat Froome and Porte but I knew that I could do well."
Britain's Adam Yates, who rides for Australian-based team Orica GreenEdge, finished eighth, 31 seconds behind Contador.
The three-week Tour de France starts on 2 July at Mont Saint-Michel in Normandy.
Stage one result:
1.  Alberto Contador (Spa/Tinkoff)               11mins 36secs
2.  Richie Porte (Aus/BMC Racing)            +6secs
3.  Chris Froome (GB/Team Sky)                +13secs
4.  Daniel Martin (Ire/Etixx - Quick-Step)     +21secs
5.  Julian Alaphilippe (Fra/Etixx - Quick-Step) +24secs
6.  Wout Poels (Ned/Team Sky)              +25secs
7.  Romain Bardet (Fra/AG2R)                    +29secs
8.  Adam Yates (GB/Orica)                     +31secs
9.  Diego Rosa (Ita/Astana)                      +37secs
10. Jesus Herrada (Spa/Movistar)                 +39secs

The mile-long Northern Inner Distributor Road (NIDR) now links the east of the town with the west.
It was estimated in January that technical issues involving a bridge increased the cost of the £21m project by a further £10m.
The council is in a legal dispute with the contractor Carillion about the extra costs and who should pay.
Somerset County Council said the delays had been "enormously frustrating".
Cabinet member for highways, John Woodman, said the new road will "unlock" the old cattle market Firepool site for development and ease traffic in the town.
He added: "The dispute is ongoing and we will work hard to make the best result for all the people of Somerset and the people of Taunton."
Mike Rigby, independent councillor for Bishops Lydeard, said the council had "a lot of questions" to answer about how the project was managed.
"How has it been that the council signed a contract that enabled a contractor to take more than twice the amount of time to build what should be a fairly straight-forward job?
"It worries me, it's [an] example of how the county council has not well-managed a large contract."
Carillion had nothing to add to the council statement.
Spain's Alberto Contador opened up a 13-second lead over main rival Chris Froome in winning the first stage of the Criterium du Dauphine.
[NEXT_CONCEPT]
A multi-million pound road scheme to ease congestion in Taunton has opened more than two years behind schedule.