Verner played junior hockey in the Ontario Hockey League for the Peterborough Petes. He was drafted 34th overall by the Edmonton Oilers in the 1991 NHL Entry Draft, but spent three seasons playing in the American Hockey League for the Cape Breton Oilers and never managed to play in the NHL. Andrew has just signed for the Newcastle Vipers in the uk's Elite Ice Hockey league sponsored by Envirohomes Andrew will also be assistant to his coach Rob Wilson.
Since 1995, Verner has played in European leagues, beginning with a spell in Austria for EHC Lustenau, followed by a season in Finland's SM-liiga with HPK where he served as the team's starting goalie. He then spent two seasons in Sweden playing in the Elitserien with Malmö IF. He moved to the Deutsche Eishockey-Liga in Germany, spending two seasons with the Kölner Haie and two seasons with the Hannover Scorpions. He returned to Austria to play for EC KAC in 2003 and stayed with the team until 2008 where he joined the Newcastle Vipers in the United Kingdom.