NFL 2012 Overtime Procedures
NFL 2012 Official CaseBook
Rule 16: Overtime Procedures, Article 3, Note 1 states:
If the second team loses possession by an interception or fumble, the down will be permitted to run to its conclusion, but if the second team's possession has legally ended with the fumble recovery or interception, any subsequent action will not affect the outcome of the game. (If the change of possession occurs in the second team's end zone, the score counts.)
Therefore, the play is allowed to continue, but the game is effectively over at the instant that the defender gains control of the ball after a fumble recovery/interception. So if Team A kicks a field goal, and Team B then throws an interception, it does not matter if the intercepting player falls down, runs out of bounds, tries to run for a TD, fumbles the ball back to the other team (who may or may not return that fumble into the endzone on the same play). Once the defensive player gains control of the ball, that team has won the game (even through the refs will not whistle the play dead). The one wrinkle is that if the ball is turned over in the endzone, the score counts, since it occured at the same instant as the change in possesion. So a team could kick a field goal, then get a turnover in the endzone to win an OT game by 9 points (since there are no extra points in OT).
So to answer the original question:
The score is tied 27-27 after regulation. The Saints win the toss, drive down and kick a field goal to make it 30-27.
If, on the Eagles' ensuing possession, Vick throws an interception, can the Saints run it back for a TD and officially win 36-27?
If the interception occurs inside the endzone, the game ends and the Saints win 36-27.
If the interception occurs outside the endzone, the game ends and the Saints win 30-27.