They didn't play well against Rams either, IIRC. Lee was about the only one who played well against the Rams on defense.
That was primarily a scheme issue.
The Cowboys were going to fail against the specific blocking the Rams used just based on the scheme (defensive alignment).
The Patriots changed their defensive alignment for the Super Bowl and the Rams blocking scheme that the Rams had rolled out against the Cowboys.
The Rams blocking scheme was designed to box out the LBs horizontally.
The Rams would what appeared to be an outside zone run but was actually a designed cut back to the inside run.
They used multiple methods to get each LB to that LBs outside and then they used delayed blocks by OLihemen to get between the LBs and the middle of the field leaving a huge space for the RB to run.
The Rams would also do the opposite in that they would get the LBs flowing in one direction and then when the ball went to the outside in the opposite direction they would have a blocker preventing the play side LB from moving to the outside.
The Patriots moved the OLB up to the line and outside of the RDE.
They flipped the Nickel CB to the other side to make up for the OLB moving to this location.
They had the RDE play the B gap (inside the LT) instead of playing the C gap like the Cowboys did with their RDE.
The OLB was there to play the C gap.
It made them a little vulnerable against the pass (in theory) but on pass plays the OLB would played a short zone and a Safety would pickup the receiver when he passed the OLBs short zone area.
That put the Nickel CB on the TE which is a good match-up on pass plays.
The end result was that the Rams could NOT cut off the path of the LBs horizontally because one stayed inside and one stayed outside with double DBs playing the run on the opposite side.
Long Story, but the Cowboys LBs really didn't have much chance against the Rams blocking scheme from the alignment they were playing and it would require too many changes to implement during a game with no practice on those changes.