For "cinematic" openings, I'd have to say Saints Row 3 and to almost the same extent, 4, has a crazy, action opening if ever there was one. The sequence has the player getting into mischief on a plane, some intense shooting and action, freefalling from high altitude and parachuting in to save the lady. It sets up an important character's role in the sequel as well. SR4, not including the tutorial/stealth mission (which itself is pretty good and sees the return of an antagonist from SR3), sees the player as the US President about to give a speech. You walk down a corridor, high fiving or slapping people as you choose. Then an alien invasion happens, Saints are being kidnapped around you so you jump onto an Uncle Sam turret and proceed to defend life and liberty with a rain of large caliber bullets
FC3 also has a great opening. While backpacking and partying around SE Asia, Jason Brody and his group find arrive on a pacific island, only to be kidnapped by pirates and held for ransom. Jason's brother creates an opportunity to escape but doesn't make it, leaving Jason alone in a tropical jungle, injured and on the run. It sets everything up really well and is quite a tense sequence.
For less drama, I think Fallout 3 and NV are both great. FO3 sees you experience your birth, being a toddler and teen in a vault, celebrate your birthday party and then stuff happens and you have to find Liam Neeson. FNV has the player wake up from a bullet to the head, amnesiac and bandaged up. The doctor who finds you asks you who you are and offers to reconstruct your face, a very clever in-game way of introducing character creation. When you can finally stand, you have a casino chip in your pocket and nothing else of note.
Lastly, I'll offer The Saboteur. As Sean Devlin, the player is the driver for a French family owned race team. They travel to Germany for a race and after some trash talk in the bar the player starts the race. Taking first on the way to the chequered flag the local favourite, Aryan specimen Kurt Dierker resorts to cheating to take 1st for the Fatherland and in revenge, Sean and his friend/mechanic/adoptive brother decide to trash Dierker's race car. They get caught in the act however and are tortured on suspicion of spying. Devlin is able to fight his way to freedom and a getaway, racing back to the farm on the outskirts of Paris. As he drives, he sees mere moments ahead of the Nazi invasion force, arriving back alive but alone. It is exciting, fun and great storytelling, setting up many characters and the occupation of Paris. Phenomenal game too, sadly Pandemic's last before EA "EAd" them.