Let me offer you an alternative way to interpret first downs and total yards....

avg starting field position- What was the average distance from the goal line for each team at the start of their drive? This can affect total yards, and also total 1st downs available. The team who wins the field position battle has shorter fields. Less yardage to gain, less 1st downs to get.
Total plays- At what pace was each offense being ran? this can affect total yards. Faster pace = more possessions = more plays = more yards. 500 yards on 100 plays < 350 yards on 60 plays. If you have more yards on more plays, you haven't done anything better than the other team. You simply ran more plays than the other team (which can be influenced by a multitude of factors)
Yards per play- Who was more efficient each time they snapped the ball? This is a much better indicator of who played better than total yards. It also has an affect on first downs. A team getting more "chunk yardage" plays eliminates the amount of possible 1st downs, compared to the team gaining a smaller amount of yardage on each play. A 30 yard play counts as 1 1st down, 3 10 yard plays = 3.

Go back and look at some box scores, and notice the overall lack of correlation between winning and total yards (unless there is a several 100 yard advantage for one side). Then calculate yards per play in each game, and note the difference in W correlation. Yards per play >>>>>>> total yards and it isn't even close.