The last time they reached this stage was when Lakers legend Kobe Bryant led them to the 2010 NBA title.
Bryant, 41, and his daughter Gianna, 13, were among nine people killed in a helicopter crash in January.
James, who joined the Lakers in 2018, said it was an "honour" play for the team and continue the "legacy".
Five-time NBA champion Bryant spent his entire 20-year career with the Lakers and is the only player in league history to have two shirt numbers retired with the same team after wearing both for a decade.
"The reason I wanted to be a part of this franchise is to take them back to a place they were accustomed to being," said three-time NBA champion James, 35. "And that's competing for a championship.
"It's an honour for me to wear the purple and gold and for us to try to continue the legacy and just play great basketball for our fans."
Kyle Kuzma scored 17 points for the Lakers, Markieff Morris got 16, Danny Green contributed 14 and Anthony Davis 13, while James Harden finished with 30 points for the Rockets.
The Lakers, who won the series 4-1, will face either the Los Angeles Clippers or the Denver Nuggets in the next round, with the Clippers leading that series 3-2.