Wing scored the only goal of the night with a fantastic drilled shot from distance that secured Boro's progression into the Cup's quarter-finals where they will face Burton Albion.
His goal always seemed likely as Middlesbrough got off to a bright start at the Riverside, breaking into Palace's box several times within the first few minutes as Eagles shot-stopper Vicente Guaita found himself repeatedly collecting the ball sooner than he'd have hoped to.
Boro appeared to have been gifted a golden opportunity when Martin Braithwaite set himself up on the end of a Paddy McNair cross only to find himself unable to bring the ball down in time to strike at the Eagles' net.
Minutes later, as the chilled Teesside atmosphere built, captain Grant Leadbitter looked to threaten the Palace goal but sent a well-drilled effort over the crossbar from the edge of the penalty box.
Boro continued to threaten and Braithwaite would have found himself through on goal had it not been for a trademark Aaron Wan-Bissaka tackle to dispossess the Danish forward.
The game had become more evenly weighted 20 minutes in, and neither side seriously threatened the other. Palace's best chance of the half came when Andros Townsend sent a lofted ball over the Boro box from close to the byline only to see his cross not capitalised upon.
Shortly after, Jeffrey Schlupp ran powerfully into the opposition's box but was frustrated to see his effort deflected into the hands of a diving Dimitrios Konstantopoulos.
Schlupp, in his second attempt on the home side's goal moments later, again found himself testing Konstantopoulos but struck straight at the 'keeper who held the ball well.
However it was Boro who came closest in the first 30 minutes as Jordan Hugill shot inches over from close-range following a probing ball by Daniel Ayala. Shortly after, Braithwaite stormed into the Palace box only to head wide from distance having gotten his head to McNair's inswinging cross.
Hugill completed a trio of Boro chances in quick succession when he shot over Guaita's crossbar.
Jordan Ayew had a brief chance to take the lead for the Eagles, but was swiftly challenged by two home players to nullify his threat. However, it was Middlesbrough who would celebrate as the second half drew to a close when Wing rifled the ball into the top left of the visitor's net from distance to break the deadlock.
Palace started the second half a much improved side, and, having switched to a 4-3-3 lineup, were able to drive their way into the home side's box from the off. Max Meyer saw a shot sail close to the post in the 47th minute as the Eagles appeared rejuvenated.
Ayew again found his attempts on goal well dealt with as his shot from close range was, much like before, blocked by a resolute home defence.
At the hour mark, the game's second half was still yet to yield a goal but it seemed that Palace were the more likely side to score it as they found themselves threatening in the Boro box continually, only to see their pressure result in nothing.
Despite Palace's apparent initial spell of dominance, the game had settled down by the 70th minute and Boro saw their only chance of note rise over the bar from a speculative Saville effort.
At the other end of the pitch, Meyer couldn't take advantage of a fine cross so instead watched his scuffed shot scooped up by a thankful Konstantopoulos while Boro found themselves ruing Fletcher's missed header minutes later.
Aaron Wan-Bissaka forced the home side into holding their breath as he consecutively threaded in two exceptional balls only for a dense Boro defence to do as they had all night and nullify Palace's onslaught with numbers in the box. Townsend came close with only minutes of the game left, but couldn't bury his effort past a strong home 'keeper.
The assault from Palace didn't ease up for the remainder of the match but, by the end of the night, it was only Wing's exceptional goal which separated the sides and Middlesbrough finished eventual winners.
Palace: Guaita, Souare (Woods 78), Riedewald, Kelly, Wan-Bissaka, Schlupp, Puncheon (Kaikai 84), Meyer, Townsend, Sørloth, Ayew (Milivojevic 65). Unused subs: Speroni, Kirby, Williams, Wickham.
Middlesbrough: Konstantopoulos, McNair, Ayala, Batth, Friend, McQueen (Saville 29), Leadbitter, Wing, Tavernier, Braithwaite (Fletcher 65), Hugill (Gestede 56). Unused subs: Lonergan, Wood, Stubbs, Chapman.