The Nomads took the last Europa League spot in the Welsh Premier League with a play-off win over Airbus in May.
Stabaek and India international keeper Gurpreet Singh Sandhu was forced into a fine save from Michael Wilde's shot.
Nicolai Naess came close for Stabaek late on with a wayward effort but Connah's Quay held on to seal a draw.
The two teams will play their reverse fixture in Norway on Thursday, 7 July 19:00 BST.
Gap Connah's Quay played out a goalless draw with Norwegian side Stabaek in their Europa League first qualifying round first leg at Rhyl's Belle Vue.