The midfielder had not featured since mid-September due to injury and he made up for lost time by firing in his first goal for the midlands club 19 minutes from time.
Leeds had taken an early lead through Pontus Jansson's header but they could not match their first-half intensity and had to settle for a point.
United had earned a much-needed victory over Middlesbrough last time out on home soil and they made a confident start, Spanish playmaker Samuel Saiz seeing an effort blocked on the edge of the area.
In-form winger Albert Adomah had a left-footed shot comfortably saved by Felix Wiedwald early on but that was as close as Villa went during a first half controlled by the home side.
Leeds were the only Championship team yet to claim a single point from a losing position and the relief was palpable when Jansson nodded in Pablo Hernandez's inswinging corner at the near post on 19 minutes.
Caleb Ekuban's driving run won United the corner and the striker was poised to pull the trigger soon after, only for Villa to scramble Saiz's cut-back clear.
Glenn Whelan shot wide as Villa briefly threatened but Leeds ended the half on top with Liam Cooper heading over Gjanni Alioski's free-kick in stoppage time.
United suffered a blow just before the break when Hernandez hobbled off and they were almost pegged back after the restart, Wiedwald saving Adomah's strike with his legs after he had found space in the box.
Saiz's ball across goal evaded three Leeds attackers at the other end and the hosts thought they had got the second goal they craved when Cooper headed home, only to be denied by the linesman's flag.
Lansbury and Jack Grealish were introduced by Villa boss Steve Bruce and the changes had the desired effect.
Neil Taylor and Jansson came together in a crunching tackle and the ball rolled into the path of Lansbury, who drove forward to fire a 20-yard shot in off the far post.
Jansson stayed down following the challenge but was fit to continue and help Leeds stem the tide.
The Swedish centre-back took a booking to halt a Villa attack during an ominous period for a United side that recently lost 2-1 from a winning position at home to Derby.
But Leeds saw out the remaining minutes without further alarm as they recorded their first draw since mid-August to move up to seventh in the table.