Marriott, who recently agreed a short-term contract with the Shakers, had to be at his best in the opening 45 minutes when Shrewsbury threatened to gain a rare away victory.
Darren Moss, Sam Aiston and Duane Darby, the latter later to limp off with a groin injury, were all foiled by the much travelled goalie.
And in the second half, Marriott was on his toes to palm away a deflected effort from his own team-mate Colin Woodthorpe. Jake Sedgemore wasn't far wide with a 20-yard free-kick for the visitors, who themselves eventually ran out of ideas.
It was hardly surprising Bury lacked a killer punch in front of goal. Jon Newby has netted only twice this season while teenager Colin Kazim-Richards was making his full debut.
In addition, Bury gave a debut to on-loan midfielder Gary Harkins, who only met his new team-mates prior to kick-off.
Bury's best effort came in the first period when a Brian Barry-Murphy header was tipped away by Scott Howie, while Dave Challinor ballooned his left-foot attempt over the crossbar.
Manager Graham Barrow said: "Although it was a patched up team I was just a bit disappointed because we can play better than that.
"However, it was nice to stop the rot and to keep a clean sheet. Gary (Harkins) looks like he has something to offer and I was pleased with his performance.
"But it says a lot that Andy was probably our best player."
Opposite number Gary Peters said: "I was satisfied with the performance, but not the score. A 0-0 draw seems like a good result away from home, but we didn't come here for a point.
"We're trying to win games playing open football, but it was a scrappy affair and I'm sure Bury will be happy with the point."