Killie lost to Stenhouseumir in the second round last season at the first stage of their defence of the trophy.
Allan Johnston's side have yet to win in the Scottish Premiership this season and Tesselaar is determined to change that record against the Scottish Championship leaders.
The Dutchman said: "We definitely need a win. Two years ago they won the League Cup and last season we had a very bad start and lost the first game against Stenhousemuir.
"So the supporters deserve a win."
Tesselaar joined Killie just after their trophy success and his former club St Mirren replaced his new team as League Cup winners last season.
So the left-back is out to create his own Hampden memories rather than hearing about them second hand.
"I never watched the cup game (final)," he said. "I just heard they won the cup. I'm happy for the boys at St Mirren. I made my own choice and have no regrets.
"Every football player wants to win a cup."
Hamilton have a perfect league record this season but have some selection issues for tonight's game.
Player-manager Alex Neil picked up a knock in Saturday's 2-0 success over Queen of the South while Jon Routledge and Louis Longridge are definitely out through injury with striker Mickael Antoine-Curier not yet available due to a delay in international clearance.
Neil said: "I will undergo a fitness test close to kick off and pick the starting XI after that. We are travelling to Kilmarnock pleased with how our season is going although we recognise that this is a step up."