Actually Jesus raised the standard even higher, by telling us that people who only follow the rules by external actions aren't doing enough-and are hypocritical. We must be clean on the inside first He tells us, then the outside follows suit, obeying for the right reasons. And the only way to become clean on the inside is to turn to God in humility, and He does the cleansing, the forgiving, the making new creations of us. Only with the help of grace can we become people who love as we should, and then love fulfills the law and excludes sin and does good for others by its nature. Paul also knew that love was the central aspect of man's righteousness and of the Christian faith; read 1 Cor 13, for one. Also Rom 13:8-10.
A major link between Paul and Jesus is that one must believe in the Father and the Son and their promises to us, and Jesus came to reveal God to an extent that He'd never been revealed before, a God we could truly believe in, who was fully worthy of our faith. And so we must know Him in order to believe, so we can then place our hope and trust in Him and ultimately love Him. This is the object of faith.
"Now this is eternal life: that they know you, the only true God, and Jesus Christ, whom you have sent." John 17:3
Faith is the doorway to righteousness (it justifies us) because faith is the doorway to God who is our righteousness, and who, alone, can make us just or righteous. Jesus didn't come to abolish either obedience or the Law (He even tells us so in Matt 5:17), rather He came to show and bring us to a place where we'll obey for the right reason. A quote from Basil of Cesarea, a 4th century bishop, is related to this:
"If we turn away from evil out of fear of punishment, we are in the position of slaves. If we pursue the enticement of wages, . . . we resemble mercenaries. Finally if we obey for the sake of the good itself and out of love for him who commands . . . we are in the position of children."