Matthews showed dominance in the sprint to easily account for the peloton to race to the third step of the podium, but the Australian was left ruing the waste of his good form.
"Everything was going really well," said Matthews. "I got to the Cipressa and the Poggio in great shape, the team delivered me really well to the climbs. "I had super legs like I was dreaming about on the Poggio. I went to attack and got put into the wall, lost most of the skin off my hand.'
The collision didn't show up on the broadcast, but it prevented Matthews from being in a position to respond to Alaphilippe and eventual winner van Aert when the pair attacked clear of the main peloton.
The two in-form riders stayed away to contest the victory, with Matthews having to settle for winning the bunch sprint to secure the final spot on the podium.
"In the end, to win the sprint against the sprinters that were here today I have to be happy with this," said Matthews. "I think if someone told me that I'd be one the podium, just not sure what step it is, at the start of the day, I'd have been happy. With the legs I had today, more was possible.
"It's a little bit bittersweet, but at the end of a hard day, to still continue after this, it shows that I just wanted to give it everything for the team today for the effort they put in for me."
"Just see what I had in the finish and it was good enough for third. I think the team will be happy with this."
Matthews is taking a different tack to the changed 2020 calendar with the sprinter and classics star not included in the Tour de France squad by his Sunweb team, and instead having the Giro d'Italia as his focus.