Dutch youngster Van Peer has emotionally battled against dartitis to win through his group in the £450,000 tournament, defeating Simon Whitlock and Cameron Menzies to reach the last 16.
However he saw his run ended as Suljovic - who had given the 21-year-old advice on how to cope with the condition earlier in the event - proved too strong with a high-quality display.
Suljovic claimed the game's opening seven legs without reply as he capitalised on four missed doubles from Van Peer, took out finishes of 110, 90 and 80 plus an 11-darter to pull clear.
Van Peer checked out 96 and 119 as he hit back, but three further missed doubles allowed Suljovic in to secure his first Wolverhampton quarter-final appearance with a 106 average and ten doubles from 15 attempts.
"Berry has a problem with dartitis but he is a super player and he played well," said Suljovic. "I was focused on my game and I played brilliant darts - my doubles were good and it was a perfect night.
"I'm happy that I won and I feel really good."
Lakeside Champion Glen Durrant became the first BDO player to reach the quarter-finals since 2013 as he overcame Dave Chisnall 10-8 with another solid performance.
Durrant took out a 161 finish as he claimed a 3-1 lead early on, only for missed doubles to allow Chisnall back in to square the contest following ten legs.
The Middlesbrough ace, though, found another gear to finish 87, 86 and 71 as he moved into a 9-7 lead before closing out a solid win with a 99.33 average.
"To be in the quarter-finals feels so good," said Durrant. "I'd targeted reaching the quarter-finals before the event but now I'm there I want to enjoy the moment.
"I'm flying the flag for the BDO and I'm really proud to be in the quarter-finals, but there's more to come from me. I feel so comfortable and I'm loving every second on that stage.
"I was disappointed to be five-all at the second break but I thought back to last year, when I was in the same position but then lost to Raymond van Barneveld, composed myself and came out firing. On my throw I played really well."
The second round concludes on Thursday with the remaining four matches, headlined by reigning champion Michael van Gerwen taking on Irish youngster Steve Lennon, who has reached the last 16 on his Wolverhampton debut.
"I'm feeling good and now that the format is longer it's better for me," said Van Gerwen. "I love the longer games because you're less nervous and you can settle down.
"I've played well in the groups but I can still improve and I'm confident."
Six-time winner Phil Taylor takes on Darren Webster - who won 5-0 when the pair met in last year's group stage - while 2012 champion Raymond van Barneveld faces shooting star Rob Cross.
Daryl Gurney plays two-time finalist James Wade, who bounced back from a loss in his opening group match on Saturday to qualify for the knockout stage of the tournament with a win over Taylor on Tuesday.
"I'm not playing as well as I can do but I showed in moments against Phil that I can hurt people when I get it right," said Wade. "I'm practising hard and if I do what I can do then it's an enjoyable game."
Tickets for the bwin Grand Slam of Darts are available through www.wolvescivic.co.uk or by calling 0870 320 7000.