A difficult evening looked to be in store for the Finnish side, who didn't manage a shot on target in the first half, while goalkeeper Faris Krkalic was equal to efforts from Juan Bedouret and Elison Rivas.
But Ilves grew into the contest after half-time, enjoying more of the ball as Jardell Kanga went close with a free-kick before Anton Popovitch went close soon after.
However, the Lynx were unable to find a way through, and had to settle for a 0-0 draw after Teemu Hytonen had two efforts thwarted in added time.
Knocked out in the Third Qualifying Round two seasons ago and the Second Qualifying Round last year, the Tampere side are aiming to reverse that recent trend and reach the League Phase for the first time, something their compatriots KuPS managed last season.
They have put themselves in a good position ahead of the second leg at home on 16th July, but will be wary that last season's runners-up in the Luxembourg league got to the Play-Off round 12 months ago.