A. T. Robertson, the great Greek scholar, has strongly stated that "His" of "His power" must refer to our Lord Jesus Christ, and I am inclined to agree, especially in the light of Hebrews 1:2 and Colossians 1:16-17, not to mention John 1:1-3. Technically speaking, then, Christ upholds all things by His power as the second person of the Trinity. Having said this, Jesus makes it very clear in the Gospel of John that He does nothing on His own, but only in submission to the will of the Father.
Further, Hebrews 1:1-4 and John 1:1-3 inform us that Jesus is the "Word." Thus, whatever our Lord does is very closely related to what the Father is doing (see John 5:17-23, 30; 8:28, etc.). It was Satan who sought to tempt Jesus to act independently of the Father (Matthew 4; Luke 4), as he tempted Adam and Eve to act independently of God.