The 20-year-old Austrian was a second-half replacement for Charles N'Zogbia and became the home side's hero by scrambling the ball over the line at the second attempt after Mark Schwarzer had failed to hold a shot from Gary Gardner.
The win is Villa's first in seven Premier League games and lifts Alex McLeish's side up to 14th in the table, 11 points clear of the relegation zone.
Fulham would have been by far the happier side with a point after a game of few clear-cut chances but, after three successive wins, the Cottagers slip down to ninth place.
With nine-goal top scorer Darren Bent out injured long-term, and Robbie Keane having returned to the LA Galaxy after a successful, but brief, loan stint, putting the ball in the net is a problem for Villa just now.
They dominated in the opening 45 minutes, but for all their possession they struggled to really test visiting 'keeper Schwarzer.
Stilian Petrov, N'Zogbia and James Collins all squandered decent openings, while Schwarzer saved well from another Petrov attempt.
At the other end, Shay Given had very little to do apart from a comfortable save from an Andrew Johnson shot, as Pavel Pogrebnyak just failed to connect with a Johnson ball across the face of goal.
Russian centre-forward Pogrebnyak had scored five goals in his first three appearances for Fulham, including a hat-trick last weekend against Wolves, but he was starved of service here and endured a frustrating afternoon.
The visitors started the second half on the front foot, but fortunately for Villa, Damien Duff's teasing ball across the face of the six-yard box found no takers.
Marc Albrighton and substitute Bryan Ruiz struck the woodwork at either end, and these sides appeared ready to settle for their second goalless stalemate of the season until Weimann popped up at the death to give Villa their first victory since a 3-2 success at Wolves on January 21.
Source: AFP
Source: AFP