Town, who beat Reading in the Championship play-off final in May, have been tipped by many to make an immediate return to the second tier.
The Terriers start the season with a trip to Crystal Palace on Saturday.
"People didn't expect anything from us last season and look what we achieved," the defender told BBC Radio Leeds.
"We are more than happy for people to write us off and more than happy to be the underdogs, that's absolutely fine. But as long as we go about our business in the right way, which I'm sure we will, then I see no reason why we can't give it a good go.
"We've got a good squad and good togetherness, and all of the attributes that we had last season I think will continue into this season.
"We know it will be tougher but we see no reason why we can't stake a claim to definitely stay in the league and progress higher up it."
The last Championship play-off final winners to avoid being relegated were Crystal Palace in 2013-14.
Queens Park Rangers, Norwich City and Hull City have all gone straight back down after winning promotion at Wembley in the last three seasons.
Smith, 25, missed much of Town's pre-season preparations with a foot injury sustained in the win over the Royals but is hopeful he will be able to feature against Palace.
"I feel good. It took a while but the doctors and physios said it would take time but I was desperate to get fit as soon as possible," he added.
"It looks like I'll be available for selection so I'm absolutely delighted with that and hopefully I can lead the team out at Selhurst Park."
Premier League newcomers Huddersfield Town are "more than happy with being written off", according to captain Tommy Smith.