Villa's Charles N'Zogbia had the first chance of the match in the fourth minute, but he shot wide of the target from 12 yards out.
Visiting Fulham came out with a more positive outlook after the restart, although they did struggle to forge out clear goalscoring opportunities.
Irish winger Damien Duff went close for the Whites in the 50th minute, but his drilled effort evaded his teammates as the ball went wide of Given's goal.
Up the other end, Villa midfielder Marc Albrighton struck the crossbar with a 20-yard effort that rebounded to safety with 23 minutes remaining.
Fulham also hit the woodwork in the closing stages when John Arne Riise found Duff, who saw his effort from ten yards out bounce away to safety via the post.
Hosting Villa exerted much pressure as they searched for a winning goal and it paid off in injury time.
Gary Gardner's initial shot was blocked inside the area but substitute Andreas Weimann was on hand to score at the second attempt and seal the three points for the home side.