Roofe coolly slotted home after 55 minutes for his fourth goal against the Rams this season as the Yorkshiremen's dominance paid off.
But United, who are looking to return to the Premier League for the first time in 15 years, will feel they should be taking a bigger advantage back to Elland Road.
Roofe was denied a second goal by a brilliant Kelle Roos save moments after his opener, while Stuart Dallas and Jamie Shackleton both went close in the second half.
But they were left breathing a huge sigh of relief when referee Craig Pawson overturned a decision to give the home side a penalty with 12 minutes to go, after Jayden Bogle and Jack Harrison collided in the box.
County, who have been promoted just once in eight play-off attempts, started brightest, with Tom Lawrence curling a sixth-minute effort wide.
But United grew into the game and went close when Dallas dipped a long-range effort just over after 13 minutes.
Roofe sent an overhead kick wide in the 19th minute, before Luke Ayling failed to make the most of a back-post header three minutes later.
Kiko Casilla did brilliantly to smother the ball at the feet of Harry Wilson after he was played in by Duane Holmes, while Fikayo Tomori was lucky to escape with just a talking to from the referee following a poor two-footed tackle on Shackleton.
Dallas volleyed narrowly wide in first-half stoppage time as the visitors ended the half strongly.
Harrison then had a volley blocked by Bogle to force a 47th-minute corner as United picked up where they had left off.
And the pressure finally told when Roofe slotted home after Harrison picked him out brilliantly following a fine advantage played by the referee.
Roos pulled off a superb full-stretch save to deny Roofe a second a minute later as Derby were left rocking.
It should have been 2-0 when Pablo Hernandez totally missed the ball with the goal at his mercy following a counter-attack in the 63rd minute, while Bogle was just unable to get on the end of a fierce Scott Malone cross at the other end soon after.
Fury erupted around Pride Park with 12 minutes to go when Derby had a penalty withdrawn after the referee initially ruled Harrison had fouled Bogle, a decision made following consultation with assistant referee Eddie Smart.
Roos kept out Shackleton's volley with nine minutes to go to give Derby hope in Wednesday's return leg at Elland Road.