Tyrone Mings (obscured) scored for Aston Villa for the first time in 55 Premier League appearances since netting in their 2-0 home win over Brighton on 20 November, 2021
Tyrone Mings scored the crucial goal, his first in the league since November 2021, with a glancing header following John McGinn's excellent inswinging corner.
Fulham saw Andreas Pereira fire an acrobatic volley into the side-netting after only 30 seconds, but struggled to create further clear-cut chances.
Unai Emery's Villa are now unbeaten in 10 Premier League matches and this victory takes them above Tottenham and five points behind third-placed Newcastle United, although they have played two games more.
They have not played in Europe since losing to Rapid Vienna in the Europa League play-offs in August 2010.
For Villa, it continues a remarkable turnaround since these two sides met on 20 October, with Fulham winning 3-0 at Craven Cottage.
Then-Villa boss Steven Gerrard was sacked within an hour of the final whistle, with the club 17th in the table having won only two of their opening 11 games.
Emery came in at the start of November and the club have since won 13 of their 20 league games and were last beaten 4-2 by league leaders Arsenal in February.
But now European football could well be back at Villa Park for the first time in 13 years - and Emery's men were good value for their win.
Ollie Watkins appeared to have a good shout for an early penalty after a challenge from Tosin Adarabioyo, but neither on-pitch referee Thomas Bramall, nor video assistant Tony Harrington, felt it was worthy of a spot-kick.
Watkins also had an effort saved by Bernd Leno, Emiliano Buendia had a shot blocked and McGinn was frustrated by Leno as Villa pushed for a second goal.
With Manchester City and Manchester United, both already guaranteed European football next season, meeting in the FA Cup final, a top-seven finish will take Villa into Europe.
Fifth or sixth would take Villa into the Europa League group stages, while seventh would leave them in the Europa Conference League play-offs.
And it's all in Villa's hands.
They still have to play fellow top-eight sides Manchester United, Tottenham, Liverpool and Brighton, as well as a West Midlands derby at Wolves, in what promises to be a thrilling final five matches for their fans.
Visitors Fulham can be forgiven for being a bit envious of Villa's position.
Having won the Championship in 2021-22, Marco Silva's side have been superb at the higher level and, as recently as mid-February, were sixth, but this was their fifth loss in seven matches to end their own European hopes.
Ninth-placed Fulham are still aiming for their first top-10 finish in the top-flight since 2011-12 and Silva will still be proud of how his side have performed in difficult circumstances.
They were still without top goalscorer Aleksandar Mitrovic as he serves an eight-game ban for pushing the referee during their FA Cup quarter-final loss against Manchester United in March.
The Cottagers suffered further blows as Brazilian midfielder Willian, who had been named in the starting line-up, had to pull out just before kick-off. Wales winger Harry Wilson was also substituted in the 16th minute, limping off with an injury.
-- Courtesy of BBC Sport