Overall leader Ben O'Connor, who had a horror day in the saddle on Friday, had a smoother ride on the longest stage of the tour, a 200km route between Villafranca del Bierzo and Villablino.
He will go into Sunday's 15th stage still wearing the leader's red jersey, 1 minute 21 seconds ahead of Roglic.
Saturday, however, belonged to Alpecin–Deceuninck rider Groves who added this win to his victory in stage two.
Van Aert, who was aiming for a fourth victory in this Vuelta, had his Visma-Lease a Bike team riding all day to control and recapture the breakaway, and secure a good position in the sprint.
The Belgian kicked first, as he likes to do, but in the dash to the line, it was the 25-year-old Quaeenslander who crossed the line first, just centimetres ahead of his rival.
"To win a man on man sprint against Wout is pretty awesome," Groves told Eurosport at the finish.
"I didn't hesitate. I think I did a pretty good sprint, and it felt like a bit of a drag race against Wout but in the end it was good enough to beat him.
"My third Vuelta and my sixth win, it's good to see the consistency paying off."
Van Aert still heads the green jersey classification 109 points ahead of Groves.
O'Connor dropped nearly two minutes to Roglic on Friday but had a more comfortable ride this time and almost snatched a couple of bonus seconds when Roglic suffered a puncture 14 kilometres from the finish.
His Red Bull-Bora-hansgrohe car was on hand to ensure a quick bike exchange, however, and the Slovenian was able to rejoin the peloton.
Roglic and O'Connor will both be aware that Sunday's stage 15 could well decide the outcome of the Vuelta.
It is a mountainous 142.9 kilometre ride with the riders tackling the Alto de Santo Emiliano and a double ascent of the Alto de la Colladiella before the Cuitu Negru with a vicious steep finale.
