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Striker Nicolas Jackson had earlier continued his fine scoring form for the hosts, but his first-half goal was not enough for victory because of Suzuki's costly error.
The Japan international could not hold on to Alex Iwobi's inswinging corner midway through the second half and fumbled the ball narrowly over the line for an own goal.
Villa were in good shape when Jackson got his sixth goal in all competitions in October - more than any other top-flight player.
He raced on to Johan Manzambi's pass, evaded weak challenges from David Affengruber and Timothy Castagne, and slotted past Bernd Leno.
That goal came in the 39th minute, and Villa should have gone ahead 23 minutes earlier when they were awarded a penalty.
Leon Goretzka's header was handled by Sander Berge, with referee Sam Barrott originally missing the offence, only for video assistant referee Matthew Donohue to advise him to watch it n a pitchside monitor.
Barrott changed his decision, but Villa captain John McGinn could not capitalise as his penalty bounced away off the crossbar.
When it was goalless, Fulham nearly went ahead when Suzuki made a superb one-handed save to keep out Oscar Bobb's effort.
Suzuki also denied Gonzalo Garcia in the second half, but he badly blundered for the equaliser and it ultimately proved costly - and beneficial for the visitors.
Villa have now won three, drawn three and lost three of their nine league matches and are ninth in the table, three points and six places above Fulham in 15th.
This match was the eighth and final one on a busy Saturday of Premier League action. It featured thrills and spills and nearly had a dramatic finish.
Villa defender Pau Torres twice had chances to snatch a late winner but failed to hit the target with close-range headers on both occasions.
After four games of this Premier League campaign, both of these teams were in the bottom three with one draw and three losses, and this draw maintains their recent resurgences.
Villa have now won three and drawn two of their five league matches since, while Fulham have picked up seven of a possible nine points in their past three games.
Jackson has had a fine October, with his goal following one in the 2-2 draw against Brentford, two in the 6-0 Champions League win over Fenerbahce and a double in the 2-1 win over Manchester City.
This encounter took place on Saturday evening rather than Sunday afternoon because Villa play in the Champions League at La Liga leaders Barcelona on Tuesday.
Some sides would be distracted by such a prospect, but Unai Emery's Villa started brightly, with Goretzka heading one chance over.
His next header resulted in the penalty after Berge raised his arm.
The penalty was the first awarded to Villa in the Premier League since April 2025 against Manchester City, having not received one last season. Surprisingly, Jackson did not take it, and instead it was McGinn who took responsibility, only to hit the woodwork.
Suzuki, who joined Villa in a £30m deal from Parma in August, had performed well until his error made it 1-1.
He did make a crucial save later on to deny Antonee Robinson as the game ended in a draw.
Sunday marks four years to the day since Emery began work as Villa head coach.
The team sat only three points above the relegation zone on 1 November, 2022.
Since then, Villa have finished seventh, fourth, sixth and fourth and made the Conference League semi-finals in 2023-24, the Champions League quarter-finals in 2024-25, before winning their first European trophy since 1982 with success in the Europa League last season.
Villa's attentions will quickly move on to their glamour tie at the Nou Camp, with Emery looking to secure their fourth Champions League victory in a row.