I think that the same basis requirements for salvation apply to everyone, as far as they are capable of complying. When Christ said "Be ye therefore perfect, even as your Father which is in heaven is perfect." (Matt. 5:48), He expected all those that chose to follow Him to obey to the best os their ability. But He also knew that everyone would fall short of that goal to some degree, each individual arriving to his own level. Then Christ, through His grace would make up for the rest, to the degree that each of us have qualified. Consider the following:
In Paul's epistle to the Hebrews, he gave a very simple answer to what is required for us to be saved. He said "And being made perfect, he became the author of eternal salvation unto all them that obey him;" (Hebrews 5:9) In other words, Christ only saves those that obey Him!
Assuming that by salvation you mean going to live in the kingdom of Heaven, Christ says very clearly, "Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven." (Matt. 7:21) In other words, to obey Christ and gain salvation, we must do the will of the Father! Otherwise, He goes on to say that even though we might believe and act in His name, if what we do is not according to the will of His Father, He will not recognize us. (v. 22) More than that, He will condemn us for doing so. (v. 23)
He goes on to explain the will of His Father, saying, "Therefore whosoever heareth these sayings of mine, and doeth them, I will liken him unto a wise man, which built his house upon a rock: And the rain descended, and the floods came, and the winds blew, and beat upon that house; and it fell not: for it was founded upon a rock." (v. 25, 26) In other words, it is the will of the Father that we learn and obey all that Christ says. According to His words, this is required to enter into the kingdom of heaven.
Obviously then, to believe in Him is not enough! We must learn and do. He made this clear when speaking to some Jews that believed in Him:"Then said Jesus to those Jews which believed on him, If ye continue in my word, then are ye my disciples indeed; And ye shall know the truth, and the truth shall make you free."(John 8:31-32) In other words, again to obey Him, we must learn and do all that He said, to gain an accurate understanding of the truth that we might become free [of deception].
Regarding the importance of learning the truth in order to be saved, our Saviour also said,"And this is life eternal, that they might know thee the only true God, and Jesus Christ, whom thou hast sent." (John 17:3) In other words, gaining salvation or "life eternal" requires accurate knowledge of His Father and of Him. That is why we must continue studying His word under the guidance of the Spirit of Truth!
With these things in mind, then, our Saviour was probably able to make this promise to that household because He could see into their hearts and already knew that they were among those who would apply themselves to learn and live His teachings sufficiently to merit His Grace unto salvation.