Rene Howe gave the Gulls a third-minute lead before a Mark Ellis own goal and an Evans' double sealed the game despite a late consolation from Danny Stevens.
After an emotional minute's applause for former Blades player and manager Gary Speed, Martin Ling's men raced ahead. Kevin Nicholson's searching free-kick was flicked on by Lee Mansell and top-scorer Howe smashed the ball past Steve Simonsen at his near post.
The Blades almost levelled the tie on 25 minutes but Evans' free-kick was spectacularly turned over by Bobby Olejnik. In-form Welsh striker Evans rattled the right-hand upright seven minutes after the restart before the comeback began.
Centre-half Ellis sliced Matthew Lowton's low left-footed cross into his own net on 68 minutes before Evans lashed home from the edge of the area following a strong run.
Evans slotted in to seal the win 12 minutes from time following Olejnik's poor clearance before Stevens powered home at the death.
Elsewhere, Michael Morrison and Jason Euell struck second-half goals to book Charlton a place in the third round at the expense of league rivals Carlisle.
Ryan Lowe's second-half goal sent Sheffield Wednesday into the third round as they beat Aldershot 1-0. Lowe finished with ease after being put through on goal by Chris O'Grady on 50 minutes.
FA Cup second-round proper results
Salisbury City 0-0 Grimsby Town
Stourbridge 0-3 Stevenageall ticket
Sheffield United 3-2 Torquay United
Colchester United 0-1Swindon Town
Chelmsford City 1-1 Macclesfield Town
Leyton Orient 0-1 Gillingham
Crawley Town 5-0 Redbridge
Luton Town 2-4 Cheltenham Town
Brentford 0-1 Wrexham
Bradford City 3-1 AFC Wimbledon
Shrewsbury Town 2-1 Rotherham United
Dagenham & Redbridge 1-1 Walsall
Barnet 1-3 Milton Keynes Dons
Gateshead 1-2 Tamworth
Sheffield Wednesday 1-0 Aldershot Town
Southend United 1-1 Oldham Athletic
Charlton Athletic 2-0 Carlisle United
Glossop North End 1-2 Runcorn Town
Newcastle Benfield 3-1 Deeping Rangers