Evanilson, who recorded a 9.4 Flashscore player rating, tallied a goal and an assist in the encounter as the Cherries notched their first win of the campaign.
The Brazilian also drew a last-man foul in the first half, which saw Sunderland defender Kevin Danso given his marching orders with a straight red card.
Evanilson unselfishly crossed for James Tavernier to open the scoring, before pouncing on Danso's indecisiveness and getting the Austrian sent off.
He got on the scoresheet after latching onto a short back pass and shimmying his way around the goalkeeper to slot into an empty net.
Evanilson ended the game with 11 touches in the Sunderland box, one big chance created and nine duels won.
Bournemouth are next in European action when they host Italian giants AC Milan in the Europa League on Thursday, before a Premier League visit to Manchester United on Sunday.