The Greek word in the text is variously rendered 'dominion over', 'masters', 'rules', but it literally translates as 'is lording over'. So, death has lorded it over the Lord Jesus Christ, at one particular point in time, but no longer does.
When you ask "who, (or what) held dominion over Jesus?" you know that death is not a 'who' and that there is someone behind death. The Bible identifies that someone as Satan the devil. With regard to death in the flesh, we are told that Jesus partook of flesh and blood,
"that through death he might destroy him having the power of death -
that is, the devil - and might deliver those, whoever, with fear of
death, throughout all their life, were subjects of bondage" (Hebrews
The Bible explains that in his pre-human state, Christ agreed to be sent to earth by the Father, to lower himself and to become subject to all that it means to live in the flesh, though without sin. This was a willing subjection, a willing humiliation to be unjustly put to death, in order to take away the devil's power of death. We know that was the outcome of Jesus' resurrection, for the risen Christ tells the apostle John,
"Be not afraid; I am the First and the Last., and he who is living,
and I did become dead, and lo, I am living to the ages of the ages.
Amen! And I have the keys of the hades and of the death" (Revelation
That answers the 'who', and also deals with the basis of the exercise of the power of death. Further to that, Jesus said to Pilate, "You could have no power at all against me, except it were given thee from above" (John 19:11 KJV). Pilate was allowed by God to put the Son of God to death. Pilate was a pawn in Satan's hand, Satan also being allowed by God to manipulate events, thinking he could crush the Son of God. But he only succeeded in bruising Christ's 'heel', not realising that Christ would then crush him, the serpent, in the head.
That was because the unlawful usurpation of God's sovereignty over the earth was proven to be unlawful at Golgotha. The righteousness of God was manifested there, and the vindication of the sinless Son of God was shown by his resurrection. Christ has lawful rights over the creation, being heir to all that he has made. And when the now installed King on heavenly Mount Sion asks the Father to inherit the nations (Psalm 2), that is what will happen.
In the meantime, Satan has been cast down, and confined to, earth, and Christ has the keys of death and hades. Not Satan. Christ holds them lawfully by allowing himself to be put to death unjustly. But as death can only claim sinners, it had no lawful hold over Christ. So we are told that "death came to all men because all sinned" and that "the result of one act of righteousness was justification that brings life... that through the obedience of the one man the many will be made righteous" (Romans 5:12, 18-21 NIV).
The tables have been totally turned on Satan. He has been lording it over sinners since he deceived the first couple into sin, using the power of death thereafter. He then thought to unlawfully use it over Christ, who allowed that to happen so that the legal rights of God be demonstrated once and for all. Now all power in heaven and on earth belongs to Christ the Lord, who holds the keys of death and hades (Matthew 28:18).