With just 700m and a tough kick to go, the 27-year-old Spaniard attacked fellow escapee Dries Devenyns (IAM Cycling) from the remains of the day's breakaway.
De la Cruz crossed the line 27 seconds clear of the Belgian while Moreno Moser (Cannondale-Drapac) finished in third, 33 seconds behind the winner.
The GC favourites rambled home peacefully together, two minutes and 56 seconds in arrears. This time advantage was enough to put de la Cruz in the red jersey after Quintana held a two minutes and 46 seconds lead on the Spaniard overnight.
"It was our objective to take the race lead today and if possible also to take the stage," de la Cruz said. "But on the final kilometres I didn't think of the leader's jersey I only thought of the stage win as I felt really close even though I was facing a good rider.
"He was strong but my legs felt good today and I managed to beat him to win. And to take the leader's jersey, it is really incredible."
"To lead my home race is the icing on the cake and I'll savor every moment."
Losing red all in the Movistar plan
Nairo Quintana now sits 22 seconds behind the race leader, but said it was all in the plan.
"By conceding this red jersey we're giving a turn to others - we're not worried at all," Quintana said.
"We remain strong, with confidence and a great group around us. It's a lead change in the race that favours us and we must continue doing things right, starting with tomorrow's important finish up the Lagos."
Movistar's co-leader Alejandro "Green Bullet" Valverde agreed.
"It's a perfect situation for us - conceding the leader's jersey by such a short time gap is just what we wanted. Tomorrow's stage will be a tough one; everyone looks tired after almost half the race at big pace, with difficult stages, so we will have to race calm.
"We spent quite some energy today, but who doesn't do that on a week like this."
As it happened
12 riders succeeded in their escape after just 10 kilometres raced. Comprising the likes of Clarke, Moser, breakaway regular Thomas de Gendt (Lotto Soudal), stage 3 victor Alexandre Geniez (FDJ) and de La Cruz, the break looked strong.
But with Imanol Erviti and Rory Sutherland at the wheel, Movistar kept the gap steady at four minutes and 30 seconds for 80kms from the category 2 Puerto de San Isidro to the unclassified El Padruin.
Here with just 24 kilometres and two category climbs to go, the stage leaders began battling it out for the stage. Several attacks failed but with eight kilometres to go, Devenyns (IAM Cycling) and de la Cruz took advantage of poor organisation behind and quickly built a lead of 20 seconds.
De Gendt took the maximum KOM points on each of the first three categorised climbs. The Belgian could not keep up with the lead group after this point but had done enough to grab the blue polka dot jersey.