The Spanish side, who have won their first two La Liga matches, had the bulk of possession and pressure and hit the bar midway through the second half, although it was far from purely one-way traffic.
Valencia's Daniel Parejo in the centre and more right-sided Roberto Soldado caused Genk's defence some difficulties, notably when the former forced keeper Laszlo Koteles into a reflex save from Soldado's cross in the 59th minute.
Valencia's Adil Rami also hit the bar in the 72nd minute with an overhead-kick.
Genk, whose coach Mario Been arrived at the club just two weeks ago, defended capably and threatened more in the second half. Defender Jeroen Simaeys forced a good save from Valencia keeper Diego Alves six minutes from time.
The result, with a first point, drew a huge cheer from the capacity home crowd, who spent the minutes after the match jumping up and down in time with music.
Genk's Simaeys was similarly upbeat.
"They had more possession than us... they have fantastic players. It was not easy... I am happy with our defensive performance today and I'm pleased with the 0-0," he said.
Been said he too was pleased with the 11-man effort and the opening of Genk's account.
"This is something you can be proud of - if you can get a result like this against a top team such as Valencia... we had our chances, they theirs. It was an open match. We are happy with this nice point."
"I've seen a very good Genk side, that dared to play and that gave little away."
The Belgian champions, now just mid-table in the league, remain without a win in the Champions League. Genk failed to win in their only previous Champions League appearances nine seasons ago and only managed one home goal - albeit in a 1-1 draw with Real Madrid - finishing bottom of their group.
In Group E's other match, Chelsea beat Bayer Leverkusen 2-0.
