John Carver has been placed in temporary charge following Speed's appointment as the new Wales manager and saw a spirited display from his charges at Bramall Lane.
Evans struck just before the break to give the Blades three vital points in their bid to avoid relegation while Swansea's promotion hopes took a blow.
Brendan Rodgers' side were looking to move into second place in the table but faced the unusual scenario of only being backed by around 50 away fans at Bramall Lane.
A combination of treacherous snow and a flurry of accidents on the motorways meant that both official supporters' coaches were turned around by the police in the midlands, while a number of Swansea officials were also forced to miss the game.
Those who did make the trip had little to see before Blades winger Stephen Quinn had the ball in the net in the 10th minute, only for the goal to rightly be ruled out for offside.
Sunderland loanee Andy Reid was looking like an influential figure for the home side whenever he was able to get on the ball, and he went close with a deflected free-kick as United pushed on.
Swansea were looking for Scott Sinclair to add some inspiration on their rare attacks and he looked dangerous when he cut in from wide on the left but could not find any power in his shot to trouble Steve Simonsen in the goal.
A poor afternoon was then illuminated by a moment of quality as the home side took the lead a minute before the break.
Reid provided it with a precise through pass which played Evans clean through on goal and the former Manchester City man clipped an early shot wide of Dorus De Vries and into the corner.
Rodgers made a double change at the interval, with Craig Beattie and Mark Gower replaced by Jermaine Easter and Darren Pratley, and Easter was soon involved in an opening.
The on-loan MK Dons forward worked a neat one-two with Nathan Dyer before flashing the ball across the face of goal and forcing Kyle Bartley into a fine clearance.
Sinclair arrowed a long-range shot into the midriff of Simonsen as the Swans looked to claw their way back level and Pratley flicked a header wide.
Swansea were dominating possession but the hosts almost made the game safe with three minutes to go.
Substitute Jamie Ward sprinted clear down the right and found Evans inside the area. The striker managed to turn and get in a shot which De Vries smothered at his near post.