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.
Andy Reid hit a precise through-ball 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.
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.
Source: PA
Source: PA