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
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.