The visitors dominated the first half and scored the only goal of the game with a trademark quick break, which James Milner started and finished himself.
Rovers rallied after the break, with Nikola Kalinic twice striking the woodwork in the space of ten minutes, but it is Martin O'Neill's men who hold a slender advantage ahead of the return leg at Villa Park next Wednesday evening.
As expected, Rovers boss Sam Allardyce made a number of changes to the side that was thumped 4-1 by Manchester City on Monday night.
Kalinic was preferred to Franco Di Santo up front, while Michel Salgado and Pascal Chimbonda replaced Keith Andrews and Gael Givet, who both failed to make the bench.
O'Neill kept faith with Carling Cup specialist Brad Guzan in goal, while Emile Heskey partnered Gabriel Agbonlahor in attack.
Rovers almost opened the scoring with little over three minutes on the clock, but David Dunn's deflected left-foot strike whistled inches past the post.
The visitors, however, who sit 14 points above Rovers in the table, soon started to take control.
They mounted a sustained spell of pressure, with countless corners, but Rovers defended stoutly.
Agbonlahor registered Villa's first effort at goal - a shot from the edge of the box on 21 minutes - but Paul Robinson made a comfortable save.
Just two minutes later however and the visitors took the lead. With Rovers pushing bodies forward on the attack, Milner broke quickly from deep inside his own half.
He fed Stewart Downing on the right and then swept home the return ball at the near post.
Villa thought they'd won a penalty on 39 minutes when Agbonlahor went down in the box under a challenge by Chris Samba, but referee Mark Clattenburg booked the England striker for diving.
The visitors almost doubled their advantage on the stroke of half-time when Downing played the ball over the top to Stiliyan Petrov, who forced a fine save from Robinson down at his near post.
Rovers, who are on a nine-game run without a win in the league, had offered little as an attacking force in the opening period and were booed off the field at the break.
Villa again threatened the Rovers goal just five minutes into the second half, but Heskey failed to make a firm connection with Ashley Young's tantalising cross.
Rovers went even closer at the other end, as Kalinic saw his free header bounce back off the post from Morten Gamst Pedersen's inviting cross.
The home side started to believe that they could get back into the tie and created another neat move moments later, which ended with Chimbonda teeing up Dunn, who volleyed well wide.
Chimbonda was the provider again on 65 minutes, cutting the ball back to Kalinic on the edge of the box, but the Croatian's crisp strike cannoned off the foot of the same post he had hit earlier.
As Rovers pushed for the equaliser, Villa almost stuck on the break, but Ashley Young fired wide of the far post after latching on to Samba's misplaced header.
Brett Emerton then tried his luck from long range, but Guzan palmed the effort around the post.
Agbonlahor had a header saved by Robinson at one end, while Steven N'Zonzi headed over from a corner at the other, as Villa, who knocked Rovers out of the FA Cup earlier in the month, remained on course to do the double over the Lancashire club by securing a hard-fought win.