The 33-year-old "Whirlwind", who has been runner-up at the Crucible six times, said: "My game has been a nightmare and I am just delighted to have got through this match. Hopefully now I can enjoy the rest of the tournament."
White led 9-6, but Henderson pulled back with breaks of 66, 50 and 58 to set up a tense decider, and the Scot had a couple of half-chances to complete a 10-9 victory. However, he missed a green early on and broke down again on the yellow with the score at 25-13. White then laid a magnificent snooker behind the blue to open the way for a winning clearance of 56.
Ronnie O'Sullivan produced a crowd-pleasing display, without the aid of any left-handed shots. He was on his best behaviour for the opening session of his second-round match against Tony Drago after being criticised for being "disrespectful" by Alain Robidoux in the previous round. He holds a 6-2 advantage going into this morning's session.