Siegrist could be facing a lengthy spell on the sidelines after he went down with a knee injury with nobody near him in training on Saturday.
United had a top-quality replacement waiting in the wings and former Motherwell goalkeeper Carson kept a clean sheet on his debut in Sunday's 1-0 cinch Premiership victory over St Johnstone in Perth.
The Northern Ireland international made a good stop from Glenn Middleton to protect United's lead after goalscorer Peter Pawlett was sent off.
The 33-year-old was the subject of a failed bid by Celtic in 2018 during the season in which he helped Motherwell reach two cup finals, before illness, the form of Mark Gillespie and then injury delivered major blows to his progress.
Smith was left in no doubt about his experience and quality after playing in front of him for the first time.
"It was seamless," the right-back said. "Benji was a massive player for us, but he's picked up an unfortunate injury and Trevor has come in.
"He's an international goalkeeper and you could see his quality in the end with a big save.
"Benji would be a big miss to any team, but I am very confident we will have no problems with Trevor coming in as he showed out there.
"He's experienced. He's been there and done it. He managed the game well at the end when we knew we needed a breather."
Smith also helped Carson out with an excellent double block during a spell of Saints pressure to make it consecutive clean sheets in the league, against champions Rangers as well as Scotland's double cup winners.
"It's something that we prided ourselves on last year when we managed to get a few clean sheets, but we struggled to stick it away at the other end," Smith said.
"We want to continue to be good defensively and that's two league clean sheets in a row. It's something that the keeper and the back four can be proud of.
"If we can keep clean sheets and add to what we have got at the top end of the pitch then it's looking good."