I see grace, salvation, and the gift as all being one who is the person of Jesus Christ.
He that has the Son, has life, and He that does not have the Son does not have life (1 John 5:12).
For Jesus was full of grace and truth (John 1:14). Jesus is the gift. For God so loved the world, that He gave His only begotten Son (John 3:16).
Anyways, believers are initially and ultimately saved by God's grace (i.e. Jesus Christ). It is why a person who comes to Jesus for the first time can have an assurance of salvation (Without doing any kind of work). Initial and ultimate salvation is a gift from God (Ephesians 2:8-9). We are not ultimately saved by works of righteousness we have done alone without God's mercy (Titus 3:5). For even if a believer were to stumble into sin, do they get clean by doing another work? No. They get clean by confessing their sins to Jesus (1 John 2:1) (1 John 1:9) and they ask Jesus to help them to overcome their sin (or to forsake it) (Matthew 26:41) (Romans 13:14) (John 15:5). But make no mistake, Scripture is also abundantly clear in that it teaches we are saved by "works of faith" after we are saved by God's grace, too.
“Ye see then how that by works a man is justified, and not by faith only." (James 2:24).
"Even so faith, if it hath not works, is dead, being alone. Yea, a man may say, Thou hast faith, and I have works: shew me thy faith without thy works, and I will shew thee my faith by my works.” (James 2:17-18).
"They profess that they know God; but in works they deny him, being abominable, and disobedient, and unto every good work reprobate." (Titus 1:16).
"If any man teach otherwise, and consent not to wholesome words, even the words of our Lord Jesus Christ, and to the doctrine which is according to godliness; He is proud, knowing nothing, "(1 Timothy 6:3-4).
"...God resisteth the proud, but giveth grace unto the humble." (James 4:6).
"And being made perfect, he became the author of eternal salvation unto all them that obey him." (Hebrews 5:9).
"Follow peace with all men, and holiness, without which no man shall see the Lord" (Hebrews 12:14).