Paterson's scrambled 61st-minute effort secured a third consecutive home win for an attritional Wednesday – their best sequence at Hillsborough since April 2017 – with caretaker boss Neil Thompson in charge following the sacking of Tony Pulis.
For Derby, the 1-0 defeat represented only the second of Wayne Rooney's 10-game interim reign and their first-half showing merited a greater reward.
The Rams spent long periods of the opening 45 minutes camped in the home half and created chance after chance without making their dominance count.
Then, moments later, Louie Sibley's stinging 35-yard drive was parried away by Joe Wildsmith and, when Kamil Jozwiak reacted smartly to pull the loose ball back to Graeme Shinnie, his goalbound attempt was turned over his own crossbar by Owls captain Barry Bannan.
Brief respite for the hosts was provided when Kadeem Harris' positive burst down the left led to an edge-of-the-box opening for Adam Reach, but his lob was always drifting too high and the Rams continued to dictate.
Midway through the half, Byrne's right-wing delivery was headed down at the far post by Sibley and Clarke's close-range prod was deflected wide off an unwitting Tom Lees.
Byrne's subsequent corner then saw Krystian Bielik's header somehow kept out by a crowd of home defenders.
Just before the half-hour mark, Craig Forsyth's pinpoint left-wing centre picked out Colin Kazim-Richards, whose diving header thudded against the Wednesday crossbar and Bielik went on to flash a 15-yard shot narrowly wide after the hosts had been unsettled by another Byrne flag-kick.
The Hillsborough outfit offered a little more attacking thrust as the half drew to a close and should have gone into the interval a goal to the good.
Bannon's free-kick from the right was curled expertly onto substitute Elias Kachunga's head but the former Germany Under-21 international somehow contrived to miss the target from four yards.
After the interval, an off-balance Sibley shot wide from 10 yards, but the Owls forged in front when Bannan's right-wing free-kick caused consternation in the visitors' box and Lees' goalbound header was flicked past David Marshall by an alert Paterson three yards from goal.
At the other end, Wildsmith grabbed Shinnie's rising 20-yard effort at the second attempt as it threatened to bounce over the line, but the hosts held on with Harris brilliantly blocking a stoppage-time Lee Buchanan shot to ensure his side make a winning start to a new year for the first time since 2016.
The victory was also only Wednesday's third over Derby during the last 25 meetings between the two sides.