Tottenham fought hard to earn a late penalty that was converted and gave them the victory.
Game was delayed by 15 minutes as visitors stuck in traffic
Townsend scored a penalty after McEveley's handball
Second leg at Bramall Lane on Wednesday, 28 January
Winners face Liverpool or Chelsea at Wembley on 1 March
Andros Townsend's penalty ensured Tottenham will go into the second leg of the Capital One Cup semi-final with an advantage over Sheffield United.
Spurs had been frustrated by the League One side before Jay McEveley's needless handball in the 74th minute gave them a spot-kick, which Townsend converted.
Emmanuel Adebayor shot over, Townsend fired wide and Eric Dier also had a header saved for Premier League Spurs.
Sheffield Utd a very good team – Pochettino
United created little and must win the second leg at Bramall Lane to advance.
Nigel Clough's side – seventh in the third tier of English football – had defeated West Ham and Southampton in earlier rounds of this season's competition but they never looked like beating Spurs.
However, the Blades had produced a resolute performance to restrict the hosts' opportunities and looked on course to head to the second leg at Bramall Lane on 28 January with a hard-earned goalless draw.
But their hopes were wrecked when defender McEveley, under pressure from substitute Roberto Soldado, raised his arms and handled in the area for a clear-cut penalty.
Tottenham have not won a trophy in the last six seasons and manager Mauricio Pochettino will be relieved with the victory, but annoyed by his side's lack of creativity.
Harry Kane, with 18 goals this season, failed to make an impact in front of watching England boss Roy Hodgson, although Kane was not the only Spurs player to be frustrated by a well-organised away defence.
Kane's strike partner Adebayor, captaining the side on his first start since 2 November, was also ineffective and wasted his only chance when he shot over the crossbar after he collected Christian Eriksen's ball over the defence.
Clough credits Blades 'work' in defeat
Adebayor's replacement Soldado also did not test goalkeeper Mark Howard, but helped force McEveley into the crucial error.
For the Blades, whose coach was stuck in London traffic causing a 15-minute delay to the match, they will be pleased to still be in the tie.
They only had 35% possession but did manage to create two notable chances.
Jamal Campbell-Ryce, who scored twice in the Blades' 3-0 win against Premier League QPR in the FA Cup earlier this month, dragged his shot wide before Stefan Scougall also narrowly shot off target.
But McEveley's mistake and Townsend's calm finish from the spot mean a draw at Bramall Lane will be enough for Tottenham to face either Liverpool or Chelsea, who drew 1-1 in their first leg, in the Wembley final on 1 March.
Sheffield United have only lost two of their 18 games in the FA Cup and League Cup under Nigel Clough
Tottenham right-back Kyle Walker began his career at Sheffield United and is a lifelong Blades supporter
Spurs striker Emmanuel Adebayor has only scored two goals in 14 games this season
Sheffield United also reached the semi-finals of the League Cup in 2003, where they lost to Liverpool
Jay McEveley's handball gave Tottenham the penalty, which Andros Townsend converted