On the last day before the race heads back into the mountains, Molano (UAE Team Emirates) was delivered by his team into the final sprint and powered across the line.
Groves (Alpecin-Deceuninck) was the favourite to win the flat 151km stage across Spain's parched plains, but things did not go to plan as he ended up in second place.
The Australian, who won back-to-back stages in the opening week, indicated he had suffered a mechanical problem.
"In the end we got bumped by UAE who came with momentum and my chain actually dropped. I managed to get it on and had a really strong sprint but unfortunately we were too far back and it was too late," Groves told Eurosport.
Molano appeared to catch the Alpecin-Deceuninck riders by surprise and out of position as he surged to the front off the wheel of team-mate Rui Oliveira.
He would have expected Groves to come back strongly but had enough of a gap to keep the Australian at bay with Boy Van Poppel (Intermarche-Circus-Wanty) coming third.
It was an uneventful day in the peloton where the overall GC battle had a day off before heating up on Friday - spending most of the stage behind the two-man breakaway of Jetse Bol (Burgos-BH) and Abel Balderstone (Caja Rural-Seguros RGA).
The duo were eventually caught, Bol resisting until being swallowed up with 38km remaining.
Surprise leader Kuss will begin stage 13 with a 26-second advantage over Marc Soler (UAE Team Emirates) and one minute and nine seconds ahead of Remco Evenepoel (Soudal Quick-Step).
Three-times Vuelta champion Primox Roglic is 1:32 back in fourth spot after earning four bonus seconds on the last intermediate sprint, with co-favourite and Jumbo Visma team-mate Jonas Vingegaard 2:22 behind in seventh place.
Friday's stage features four big climbs in the Pyrenees with the mighty Col du Tourmalet the climax.