The offense can help the defense out alot by performing better. If you get a lead it starts to limit the opposing offense playbook and make it easier to play defense against. If you are getting a lead then you are not leaving your defense on the field for long stretches, getting them tired out for when you need them late in the game. No defense holds up over the game if the offense is having short drives, leaving poor field position, and not putting up points. So whatever you think of the defense, it will require the offense to perform like a top 5 offense to win.
If the defense was tired in the 2nd half Sunday it's their own fault for allowing drives. The offense was fine in the 2nd half, and they generally (aside from the interception) possessed the ball okay in the first half too (drives of 7 minutes and almost 6 minutes). They faced four overall possesions in the 2nd half. FOUR. If they were worn down that's on them. They faced nine possessions for the whole game. That's extremely low for an NFL game.
As for getting ahead, you can't limit the other team's playbook if you can't stop them in the beginning of games. You can only do that by getting ahead by more than a score, and that requires stops. They can win with how they played defense in the 1st half (even considering NE's a very poor offense). The 2nd half not so much.
EDIT: To pound the point home: the Bills gave up 4 points per possession in the 2nd half Sunday, and the Patriots drive starts were dead on the league average - so it clearly wasn't because of field position. Additionally, the Bills had more of the ball than New England in both halves, so they shouldn't have been worn down. And to put that number in perspective, the BEST offense in the NFL currently averages 3.01 points per possession. New England, after that 2nd half explosion, remains 30th in the NFL scoring 1.26 points per possession over the full season.
They could hold significantly better offenses to that 4.25 points per possession mark (which would obviously be an improvement over allowing that to one of the leagues worst offenses), and they're still going to have a boat load of trouble winning more than just a game here or there like that. IF they play like they did in the 2nd half the rest of the year, they won't get to .500 no matter how good the offense plays. They must improve.