The 171-kilometre route from Atripalda to Salerno had riders spend the majority of the ride in the rain with little relief, riding cautiously on the descents, but chaos ensued when two big crashes ahead of the final sprint split the bunch.
The first took place seven kilometres from the end, with Groves among those to fall on a bend and the second occurred with less than two km left when Evenepoel was brought down again along with many riders.
Andreas Leknessund (23) avoided both crashes and retained the leader's maglia rosa jersey with a 28-second lead over Evenepoel.
The second crash did not affect Evenepoel's time as it happened inside the three-kilometre mark, effectively neutralising the general classification times.
But the crash left the Belgian world champion fuming while the rest of the bunch surged ahead and Alpecin–Deceuninck's Groves dominated the sprint to win the stage.
"I surprised myself today. Everything was going well but then I crashed at the roundabout with seven kilometres to go," the Australian said.
"Luckily, I put my chain back on fast enough and the group came back together. The guys did a good job earlier and then luckily I was good enough to be in position and had the legs to lead out and win.
"It's a dream to win, this is a race I've been focusing on since November or December... After two third places this week, they (the team) delivered me to a win."
Stage two winner Jonathan Milan (22) and Mads Pedersen (26) were in hot pursuit but finished second and third respectively, while Mark Cavendish (37) and David Dekker (25) crashed on the finish line.
DOGGED DISPLAY
Earlier, Evenepoel hit the deck after a dog ran on to the wet road, taking him down as well as Soudal-Quick-Step teammate Davide Ballerini (28), who slid several metres on the tarmac.
But Evenepoel, wearing his rainbow jersey after losing the maglia rosa on Tuesday, was back on the saddle again two minutes later before catching up with the bunch.
After two climbs, the route flattened out with 30km to go. The breakaway trio of Thomas Champion, Samuele Zoccarato and Stefano Gandin had built up a head of steam before they were absorbed by the peloton with seven km to go.
That was when the first major crash split the bunch and prevented some sprinters from moving up for the final attack, including Movistar's Fernando Gaviria and UAE Team Emirates' Pascal Ackermann.
As the teams scrambled to bring their lead-up riders to the front with less than two km to go, another crash brought Evenepoel down and the rest of the bunch went ahead for the sprint.
Cavendish found himself jostling for position before he fell on the line, taking out Andrea Vendrame who was carried away on a stretcher.
Primoz Roglic (33), another pre-race favourite, also crashed and suffered a cut leg.
"It's all good, could be a lot worse. The guys did an incredible job to bring me back. Just super bad luck, with the second crash," he said.
