Townsend kept his cool from 12 yards after Jay McEveley was punished for handball to ensure Tottenham made home advantage count ahead of next week's return fixture at Bramall Lane.
Pochettino's side laboured for much of a cagey game against the League One promotion hopefuls, but that will matter little if Tottenham finish the job and book a place in next month's showpiece against either Chelsea or Liverpool.
Given United's giantkilling exploits over the past two seasons – knocking Aston Villa, Fulham and QPR out of the FA Cup and beating Southampton in the League Cup quarter-finals – Pochettino will be thankful that his players produced a professional display as he targets silverware in his first campaign in charge at White Hart Lane.
The standout pre-match team news revolved around Tottenham striker Emmanuel Adebayor, handed his first start since the beginning of November following a spell of compassionate leave back in Africa and even given the responsibility of being captain for the night.
When the action eventually got under way - kick-off was pushed back by 15 minutes due to local traffic problems that resulted in the United team bus arriving late at the ground - a pumped-up Adebayor received an early yellow card for a raised arm on Louis Reed after 17 minutes.
Journeyman winger Jamal Campbell-Ryce blasted a low shot across goal after cutting in from the right, while at the other end, United goalkeeper Mark Howard got a strong hand to turn behind Eric Dier's header from Townsend's free-kick.
Adebayor spurned an excellent opportunity 11 minutes before half-time when he beat the offside trap to meet Christian Eriksen's long ball, but the striker's first touch was too heavy and his effort soared over.
Tottenham were then given a scare when Ben Davies was dispossessed on the edge of his own penalty area by Reed, who squared for Stefan Scougall to shoot just wide.
On the stroke of half-time, Dier's flicked header from another dangerous Townsend set-piece had Howard scrambling across his goal-line, but it was off target.
The second half started just like the opening period, with little in the way of clear-cut chances.
Pochettino opted to replace Adebayor with Roberto Soldado just after the hour mark in an attempt to rouse a response from the hosts and it duly came as the substitute was involved in the penalty incident that changed the game.
Soldado controlled Jan Vertonghen's chipped pass near the byline and McEveley's momentum in trying to keep pace with the Spaniard saw his outstretched arm connect with the ball.
Howard guessed the right way from Townsend's resulting spot-kick, but was unable to keep it out, despite getting close to it.
Nigel Clough's United side responded well to going behind without really troubling Tottenham goalkeeper Michel Vorm, and will need a positive attitude next Wednesday if they are to mastermind another cup upset.
