Let’s say the starting pitcher gives up the tying run in the top of the 6th, and his team unties it in the bottom of the 6th. Does he need to come out for the 7th to be eligible for the win? BTW, why yes I AM watching the Red Sox game. 
As long as he finished the 6th, he’s the pitcher of record.
To put it another way, once a pitcher has become the pitcher of record, which Beckett had upon recording the third out of the top of the fifth inning, he remains the pitcher of record unless and until another pitcher qualifies.
He’s not the Pitcher of Record anymore! The 'pen (mostly Papelbon) gagged away the lead in the 9th!
Just a slight delay, they won it in the 14th. 
The relevant rule here is 10.17.
[QUOTE=Official Rules]
The official scorer shall credit as the winning pitcher that pitcher whose team assumes a lead while such pitcher is in the game, or during the inning on offense in
which such pitcher is removed from the game, and does not relinquish such lead,
unless…
[/QUOTE]
The “Unlesses” are a starting pitcher who does not pitch enough innings (five innings in a full game, four innings in a game that only goes five, or various special cases in exhibition matches) or a rule that is rarely invoked - a relief pitcher who is not effective when a subsequent relief pitcher is effective in preserving the lead.