The Champions League holder has had to adjust to life without head coach Zinedine Zidane and superstar attacker Cristiano Ronaldo as it seeks to close the domestic gap to Barcelona in LaLiga during the campaign ahead.
Facing Serie A opposition at Metlife Stadium in Rutherford, Madrid picked up where it had left off in a dominant 3-1 win over Juventus last time out.
With just two minutes gone, Gareth Bale's brilliant pass with the outside of his boot released Asensio in behind Ivan Marcano, and the attacker was left with a simple finish to beat Robin Olsen and make it 1-0.
Bale took matters into his own hands in the 15th minute, receiving a lofted pass out of defence from Dani Carvajal, cutting inside Marcano with ease and slotting past the helpless Olsen to double Madrid's lead.
Roma responded by dropping deeper but failed to mount a consistent threat of their own, Edin Dzeko forcing a save from Keylor Navas with a long-range shot in what was one of Roma's few first-half opportunities.
At the other end, Madrid continued to cause problems. Benzema forced a good save from Olsen with a fierce drive from just outside the area in the 36th minute. And two minutes later, Asensio had a shot blocked at close range after a flowing move down the right.
Sweeping changes from both sides, including the introduction of Justin Kluivert, helped the Giallorossi to rally in the second half.
Kevin Strootman duly pulled one back for Roma when he met Patrick Schick's flicked header from a long throw unmarked at the back post in the 83rd minute, in what was just reward for an improved display in the second half from the Italian side, but they proved unable to force a late equaliser.