Spaniard Soler (UAE Team Emirates) attacked from a lead group late on and stayed away to cross the line four seconds ahead of the chasers including Frenchman Molard (Groupama-FDJ).
Defending champion Primoz Roglic (Jumbo Visma) had taken the red jersey on Tuesday but finished back in the peloton and is now fifth, four minutes and nine seconds back.
It was no surprise, however, that Roglic's team relinquished the jersey as they will bide their time and the Slovenian will still be favourite to claim overall victory.
Molard, thanks to three bonus points he claimed on the final climb of Alto del Vivero, will take a slender two-second lead over Britain's Fred Wright into stage six.
Soler appeared to have been in trouble on the day's final climb having bridged across to the breakaway group, but he burst to the front with 2.5km of the ascent left.
It still looked as though he would be caught on the descent into Bilbao but he maintained a 10-second lead and the chasers left it too late to reel him in.
Thursday's stage includes the first category one climb of this year's race with a summit finish at San Miguel de Aguayo.
Earlier in the day the race had its first COVID-19 case with Daan Hoole (Trek-Segafredo) ruled out.