Actually, the Bible doesn't claim to tell us what his last words were. As we compare the different Gospels and Acts, it would appear that perhaps his last recorded words are those found in Acts 1:4-8, but the text does not say that these were his very last words. Matthew does not contain an account of the Ascension, so there is no way to know when the Great Commission was spoken (Matt. 28:18-20). Mark 16:15-18 records some final words of Jesus, but does not claim that they are his last words. Further, these verses are not even in the better Greek manuscripts of Mark. Probably, they were added later by a Christian scribe who thought the gospel needed a better ending than that provided by Mark 16:8 (the last verse of the better manuscripts). Most scholars do not consider Mark 16:9ff. to be part of the inspired text. Luke 24:51 explicitly tells us that Jesus' last words are not recorded in Luke's gospel, and John contains no account of Jesus last words or events.

Or perhaps you are thinking of his last words before he died. Matthew 27:50 explicitly states that Matthew did not record Jesus last words before he died. The same is true of Mark 15:37. Luke 23:46 suggests that Jesus last words before his death may have been "Father, into thy hands I commit my spirit," but Luke does not actually say that these were the very last words Jesus uttered. The only thing that the grammar of Luke's account tells us is that Jesus said these words before he died -- no big surprise that he would have spoken before he died rather than after. Still, the implication of the context of this verse is that these were among Jesus' final words, but that is only an implication, not an explicit claim of the text. The idea that these were precisely Jesus' very last words is, in my opinion, only a weak implication at best. If these were merely among the last words Jesus uttered, perhaps just part of Jesus' last few statements, Luke's account would not appear any less accurate. Jesus' last words before his death as recorded in John 19:30 likewise contain no explicit claim to be his very final words, or to be the full content of his last few statements. If he said both the words in John 19:30 and Luke 23:46 in the same context, the truth of neither text would be compromised. Neither explicitly claims to records Jesus' precise and very last words, and neither claims to record everything that Jesus said with his dying breaths. The Bible simply does not claim to tell us his last words before his death, just as it does not claim to tell us his last words before his Ascension.