After failing to be drafted at 18 years of age, Peverill played with the Coburg Tigers in the VFL for two seasons. He caught the eye of Melbourne selectors and for the next year played on the supplementary list. After a year at Melbourne, the Demons coaching panel decided to discard Peverill and the former Keilor Park player ended up on the Essendon top-up list for the 2000 season. He was rewarded with position on the Bombers' rookie list in 2001 and then came his big break.
Peverill's good family friend Joe Misiti unfortunately injured his knee in Round 3, 2001 against Carlton. Misiti had an arthroscope and was put on the long-term injury list. With this came an opportunity for Peverill to be promoted onto the senior list. ""Joey's old knees gave way on him and he was a chance to come back within six or seven weeks but he chose to be put off the list for eight weeks and give a rookie a chance. I was lucky to get promoted and get a game,"" Peverill said.
Peverill said he wasn't that nervous before his debut match in Round 4 against Sydney and thinks it was because it had been so long in the making. ""Before my first game I was expecting to be more nervous and more excited. It just felt like any other game. I was probably lucky that I was 21 at the time and had been around for a few years. Robbie Forster-Knight played his first game in the same match and he didn't say a word before the game because he was pretty shakey. Probably being a bit older helped me out with the nerves. I just went out there and played like I normally would.""
Peverill played a big part in the team's 47-point victory and notched up 14 kicks and 6 handballs. He was the fourth highest possession getter for the match. ""The next game was Anzac Day and I started on the bench there. Then I went on in the second quarter and did alright again and things went on from there,"" Peverill said. Peverill continued to go from strength to strength and played the next nine games. ""I was lucky to play nine reasonable games and unfortunately broke my foot and had to go back on the rookie list. Just to play nine games when I didn't think I would play any that year was a bonus for me.""
After his final game against Adelaide in Round 12, a reluctant Peverill was clapped off the ground by his teammates and the Essendon crowd in recognition of his contribution to the team. Peverill appreciated the accolade but said he didn't expect it and slightly embarrassed about the recognition. ""It is something that I will look back on in a few years or at the end of my career and be proud of what I did that year. That game against Adelaide I suppose was the end result of what I had achieved all year. Once I got over the embarrassment it was a nice touch by Sheeds and the boys,"" Peverill said.
Peverill was elevated to the senior list by Essendon at the end of 2001 after going through the pain of missing out on senior selection at seven previous drafts. ""I suppose like most blokes you just wanted to be drafted by any team and get the chance to play AFL. To finally end up at Essendon and get the chance to play for them worked out well for me because I grew up barracking for Essendon and just lived down the road. When I came to the club I already knew a lot of the guys so it definitely made it easier to fit into the club.""
