FULL TIME: Aston Villa 1 Leeds United 0 - Grabban goal sinks Whites despite improved second half
After plenty of Villa pressure, the high-flying hosts took the lead in the 29th minute when Grabban easily beat the Whites defence to a cross from Jack Grealish to place a neat header into the far corner.
But Leeds improved after the interval with the Whites guilty of squandering some good opportunities before Villa finished on the front foot only to be continually denied by the excellent Bailey Peacock-Farrell.
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Hide AdVilla began on the front foot but Grabban fired well wide in the second minute before United were only denied an opening goal in the third minute when Sam Johnstone tipped Gjanni Alioski's fierce effort over the bar after the Macedonian cut inside from the right wing.
But Villa gradually began to turn the screw and Matthew Pennington and Stuart Dallas were forced to scramble clear a dangerous cross from Robert Snodgrass.
Jonathan Kodjia then wasted an opening down the left when firing straight at Bailey Peacock-Farrell who then produced a strong save to keep out a curling effort from Snodgrass from 20 yards out.
United's fans then probably feared the worst when a foul on Snodgrass gave Villa a free-kick on the edge of the box but Snodgrass thumped his effort into the wall and ran the follow up attempt out of play.
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Hide AdUnited's Caleb Ekuban was then perhaps unlucky not to win a free-kick on the edge of the Villa box after skinning Mile Jedinak but Villa were soon back on the attack and Grealish fired just wide from the edge of the box after a corner was cleared.
But the Villa pressure was mounting and Leeds caved in one minute before the half hour mark when an over-hit cross from Snodgrass was collected on the opposite side by Grealish whose delivery was met by Grabban.
The Villa striker easily escaped any would-be markers and simply nodded home a header across Peacock-Farrell into the bottom left corner.
Villa continued to attack and Peacock-Farrell held firm to stop a curling effort from Grealish before Johsua Onomah wasted another opportunity by blazing over.
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Hide AdYoung left back Tom Pearce then tangled with Kodjia in the area but referee Tony Harrington waved away Villa's claims for a spot kick.
Despite only holding a one-goal advantage, Villa appeared to be cruising and Whites head coach Paul Heckingbottom opted to bring on star Spanish playmaker Samu Saiz for Ekuban during the half-time break.
United immediately improved after the interval and after Pablo Hernandez had blazed a wild effort over, Villa's Glenn Whelan was well placed to head a Kalvin Phillips volley off the line.
Snodgrass then fired another free-kick into the wall but United were on top and Ronaldo Vieira fizzed a powerful effort over the bar from another cleared corner.
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Hide AdBut a reminder of the Villa threat was provided just after the hour when Peacock-Farrell twice thwarted Kodjia from close range.
Leeds were soon back on the attack and Roofe took too long to shoot after a lovely cross from Saiz, instead passing to Phillips whose shot was blocked.
But Villa weathered the Whites storm and the hosts would have doubled their lead in the 76th minute but for a brilliant save from Peacock-Farrell to deny Adomah with his legs.
Pennington then produced a superb last-ditch tackle as substitute Scott Hogan looked to pull the trigger when clean through on goal.
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Hide AdLeeds then looked for one last rally in three minutes of added time but Villa had done enough to record the victory that took them within two points of second with Leeds left lumbering in 14th and heading for 15th if QPR win at home to Preston on Saturday afternoon.
Leeds United: Peacock-Farrell, Pearce, Pennington, O'Connor, Dallas, Vieira, Phillips, Alioski (Grot 66), Hernandez, Roofe (Lasogga 78), Ekuban (Saiz 45). Unused Substitutes: Lonergan, Anita, Struijk, O'Kane.
Aston Villa: Johnstone, Elmohamady, Chester, Jedinak, Taylor, Snodgrass, Whelan, Onomah (Samba 85), Grealish, Grabban (Hogan 77), Kodjia (Adomah 67). Unused Substitutes: Bunn, Bree, Lansbury, Hourihane.
Referee: Tony Harrington.
Attendance: 33,374.