Finishing on the special-category Alto de la Covatilla, Kwiatkowski was climbing at the front of a select GC group before an increase of the pace saw the Pole slip back.
De La Cruz had been riding to his own tempo and measured his performance to drop 52 seconds at the line. That moved the Spaniard up to 13th place overall at the first rest day, 1:26 back on the new race leader.
That honour belonged to Simon Yates (Mitchelton-Scott), who finished among a sprinkling of contenders, 2:49 back on breakaway winner Ben King (Dimension Data) after 200.8km.
Kwiatkowski dropped a little over two minutes in the closing kilometres and now sits 15th overall, 2:10 back on red. Yates holds a slender one-second lead over Alejandro Valverde (Movistar).
11 riders had headed up the road early on, and despite Groupama-FDJ riding on the front the gap spun out to the 10-minute mark. Overnight leader Rudy Molard was another to slip back on the final climb.
Jonathan Castroviejo had arrived at the front to set a tempo on the early slopes, while Pavel Sivakov was able to remount and finish the stage despite a crash on the approach to the final climb.