The home side were the stronger outfit for large periods of the game, particularly in the first half, and had plenty of chances but couldn’t find the back of the net.
Instead it was Mossley who had the biggest opportunity to win the game but, with just two minutes left on the clock, Ben Halfacre hit the post and Adam Morgan - of Liverpool FC fame - couldn’t convert the rebound.
Despite not getting all three points, United manager Andy Welsh said it was a “positive” start to the campaign.
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“That was exciting to watch without a doubt,” he said.
“Me and Quinny (assistant boss Paul Quinn) are saying this is an exciting team that we’re watching here.
“We’re frustrated because we had that many chances. With the missed opportunities it could have been three or four in the end.
“First game of the season you want to make an impact, and fans are saying the same: how have we not won it?”
Ossett made a bright start to the season opener with a chance inside the first minute.
Luke Porritt had a back-post volley scrambled away by Mossley on the line after right-back Ryan Toulson’s delivery.
They had further chances early on with Andy Monkhouse a whisker away from a fantastic goal as his left-footed drive from the edge of the box whistled past the post, before Damian Reeves had a shot blocked by Adam Jones.
Andy Ijiwole was United’s brightest threat and brilliant skill down the right allowed him to lift a cross to Elliott Harrison at the back post, but the midfielder could only put it over the bar.
Ijiwole curled a free kick narrowly over the top before creating another chance.
The Huddersfield Town loanee crossed for Chris Chantler whose header was blocked, with Harrison off target with the follow-up.
Mossley didn’t have a single shot on target in the first 45 minutes, which may explain why Morgan tried his luck straight from the restart following the break.
The audacious effort didn’t find the target but it did kick-off a more exciting second period.
Ossett’s best chances of the half fell to Reeves, who had a drive saved by Lilywhites keeper Dylan Forth and headed Porritt’s cross over the bar, with a composed finish in between ruled out for offside.
Both sides had to scramble to keep the score level as well, with Harrison and Reeves having efforts blocked at one end and, six minutes later, Halfcare and Ryan Brooke denied by desperate defence at the other.
The big chance in the final minutes fell to Mossley as substitute Halfacre struck the woodwork and the rebound fell to Morgan, but he somehow missed the target from point-blank range to ensure a goalless finish.