Cricket latest: Yorkshire edge opening day in Roses clash
Lyth, the top-flight's leading run scorer with 953, added his third first-class century of the summer as Lancashire struggled early on without England Lions bowler Sajid Mahmood.
The left-hander, 22, reached 100 off just 122 balls with 17 boundaries but was out two balls later to Tom Smith.
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Hide AdYorkshire closed on 379-8 from 96 overs, with stand-in captain Rudolph making 83 off 153 balls and Anthony McGrath 61 off 136.
But it should have been far better for the Tykes because they lost six wickets for 74 runs in the evening session to slump from 299-2.
Four of them went to captain Glen Chapple in the space 26 balls and he finished with impressive figures of 4-60 from 23 overs. Smith and Daren Powell also claimed two wickets apiece.
See Tuesday's Yorkshire Post for Chris Waters's first day match report.
Yorkshire: Rudolph (captain), Lyth, McGrath, Bairstow, Brophy, Rashid, Pyrah, Rafiq, Best, Patterson, Hannon-Dalby.