Cook went past Graham Gooch's record haul of 8,900 test runs for England shortly after lunch and made 75 in an opening partnership of 177 with Lyth who hit 107 in his second appearance.
New Zealand fought back strongly in the final session, however, picking up five wickets to leave the match finely poised with England 97 runs behind.
Cook and Lyth started cautiously compared with the touring side's flamboyant stroke-play that lit up Headingley on Thursday and England went to lunch on 54 for no wicket.
Left-hander Cook drove Tim Southee to the cover boundary to go past Gooch's record and immediately knuckled down in search of his 28th test century.
Lyth struck a few crisp boundaries through the off side and his fifty included seven fours.
Cook soon followed him to a half-century, carving his ninth four over the slips, and the pair looked completely in control on 163 for no wicket at tea.
Captain Cook was then trapped lbw by off-spinner Mark Craig, misjudging a straight delivery and being given out after a review.
His departure seemed to unsettle Lyth who played the ball on to his stumps without dislodging the bails on 90.
The compact left-hander spent 40 minutes in the nervous 90s before striking successive fours off Craig to reach his hundred.
Lyth celebrated wildly in front of his home Yorkshire crowd, punching the air and acknowledging the warm applause.
Gary Ballance, who failed twice in the first test, took 18 balls to get off the mark but soon began to pepper the boundary with trademark square cuts.
Lyth, looking weary, was run out when his partner called him for an impossible single and Ballance, on 29, was bowled by Trent Boult with the second new ball.
Joe Root edged Southee to wicketkeeper Luke Ronchi for one and Ben Stokes, on six, then clipped Boult to Craig at slip in the penultimate over.
Ian Bell (12 not out) and Jos Buttler (six not out) will resume in the morning.
Earlier, the New Zealand tail wagged merrily after resuming on 297 for eight, adding 53 runs in a lively half hour.
Stuart Broad took the last two wickets to complete figures of five for 109 and Craig finished unbeaten on 41 in a total of 350.
(Editing by Justin Palmer)
Share
