The stat that I saw tweeted was an attempt to explain the Patriots' success on 3rd down. The tweeter used the commonly-referenced yards-per-1st-down metric and concluded: the Patriots averaged 7.1 yards / 1st down, making 3rd down conversions manageable; hence a high conversion rate.
I think yards/per references are valuable, but only when used in controlled circumstances. For example: 1st down could lead to a variety of results: if the Patriots threw for 30 yards on a 1st down play, that would certainly influence a number of things, including their yards/1st down metric. However, a 30-yard bomb would result in another 1st down without having a 3rd down in-between; therefore, this specific play has nothing to do with third down conversion success.
So what could explain the Patriots effectiveness in extending drives? I went through the play-by-play and came up with a refinement of the stat. Instead of lumping all 1st downs together, I charted for both teams 1st downs that led to a 2nd down conversion, 1st downs that led to a 3rd down (converted or not), average 3rd down distance, and 1st downs that resulted in big plays. The results are as follows (I threw out a few garbage possessions at the end when the Texans had Yates in at QB and the Patriots were kneeling):
First, isolating the effects of big 1st down plays (10+ yards) had a big difference. Both teams had the same number of such plays and produced a similar number of yards/play, showing that these big plays did not swing the game. Also, the Patriots and Texans had a similar number of 1st downs that led to a 3rd down scenario; not surprisingly, their performance on these downs was not spectacular at sub-2.00 yards/play, which makes sense given that an unsuccessful 1st down play should lead to 3rd down. While the Patriots were slightly better on these plays (0.224 yards better), I do not believe that to be a statistically significant difference given the sample size. Basically, the Patriots and Texans were equally unsuccessful on these types of 1st downs, inferring 1st down yardage may have nothing to do with 3rd down success, which is surprising.
So what did make the difference? First, compare the first row in each chart. I wanted to isolate the effect of 1st down yardage that led to a 2nd down conversion, as this does not apply to 3rd down conversion %. The Texans had more success in terms of yardage on 1st downs leading to a 2nd down conversion (almost a full yard better), but the Patriots had TWICE as many successful 2nd down conversions. The Patriots were doing more with less on 2nd down. Also, compare the last row in the charts: average 3rd down distance. Both teams faced a similar number of 3rd downs, but the Patriots faced on average almost a full yard less in these situations. Given that the Patriots were only slightly better on 1st downs leading to a 3rd down, we can infer that the vast majority of damage was done on 2nd down.
Looking at more focused stats, even in a small sample size, I've concluded that a big reason for the Patriots success in this game was their success on 2nd down. They faced, on average, longer 2nd downs, but they converted far more of these tough downs and on the ones they didn't, they picked up sufficient yardage to make third down more manageable. 2nd down was the difference in the game.