The Sri Lankans are still 435 runs behind and need a further 235 to avoid the follow-on.
They lost opener Dimuth Karunaratne for 38 as debutant Chris Jordan claimed a wicket in his first over of Test cricket.
But Kaushal Silva added 82 runs for the second wicket with Kumar Sangakkara as the tourists ended on a positive note after a tough start. Sri Lanka haven't lost a Test Match at Lord's in 23 years.
Root became only the second Englishman to score a double hundred against Sri Lanka after Jonathan Trott and helped the hosts to their highest score against the Sri Lankans surpassing their 551 for six declared made in 2006.
Resuming from the overnight score of 344 for five, Root extended the partnership with wicketkeeper Matt Prior to 170 runs, also a record for England against Sri Lanka for the sixth wicket.
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The one-dimensional Sri Lankan attack struggled on good batting conditions as England scored