Aleksandar Mitrovic also netted a three-minute brace for Slavisa Jokanovic's side, who confidently shrugged off the disappointment of successive defeats back in the top flight to pick up all three welcome points.
Jeff Hendrick struck six minutes after Seri's spectacular early opener and a close-range James Tarkowski finish late in a five-goal first half gave Burnley hope they could yet add to the lone point they have managed in a sluggish start to the new season.
But Mitrovic scored twice with his head in between Burnley's goals and a deserved late strike for Schurrle left Sean Dyche – who knows his side are teetering on the edge of Europa League elimination after a first-leg defeat at Olympiacos on Thursday - with mounting problems to solve.
An entertaining first half began with Seri lashing a rising 25-yard drive beyond Joe Hart in the fourth minute, only for Hendrick to hastily respond by capping off a counter-attack at the second attempt.
4' OH MY WORD!!! YOU CANNOT BE SERI-OUS!! WHAT A GOAL!!! pic.twitter.com/pgiXr4KF1g— Fulham Football Club (@FulhamFC) August 26, 2018
Burnley are not the defensive force of last season, though, and were unable to stop Fulham scoring twice from crosses in quick succession, Mitrovic proving a menace with commanding headers past Hart.
The Clarets did battle back into the contest when Ben Mee rose to meet a corner that led to Tarkowski turning home amid suspicion of offside four minutes prior to the interval.
Fulham responded to the challenge by continuing to control possession after the break and Schurrle, denied earlier by the crossbar, eventually got his reward in the 83rd minute, the Germany international drilling home the rebound from Mitrovic's initial attempt to seal the result.
What it means: European exit may be best
It will be intriguing to see how Dyche approaches the second leg of the Europa League play-off tie against Olympiacos, with a 3-1 loss on the road making qualification for the group stage an unlikely prospect.
The need to rotate appears to have already come at the cost of Premier League points, and defeat on home soil this Thursday may be more readily accepted than at any other juncture.
Pat on the back: Seri a shining star
A big-name signing during a busy transfer window, Seri illustrated the ability that once captured Barcelona's attention with an all-round performance of immense quality. His spectacular goal set the tone for 90 minutes of classy touches and limitless energy.
Boot up the backside: Burnley's defence
Though more assured after the interval, the Clarets endured a disastrous first half that left them chasing their tails. The imposing Mitrovic found space too easily for his twin headers, an indictment on the partnership between Mee and Tarkowski that provided such a solid foundation for the team’s success in the league last season.
Key Opta stats:
15 - Since his debut for Fulham in February, no player has scored more league goals in the top four tiers of English football than Aleksandar Mitrovic (15, level with Mo Salah). Fire. pic.twitter.com/alpe3k9NuK— OptaJoe (@OptaJoe) August 26, 2018
- Since a run of five consecutive Premier League wins between March and April, Burnley have collected just three points in eight league games (P8 W0 D3 L5).
- Burnley goalkeeper Joe Hart has conceded at least three goals in five of his last six Premier League games in the month of August, conceding 17 goals in total.
- Andre Schurrle scored his first Premier League goal for Fulham – his goal was the 11th shot he’d had in the match.
- After failing to score with any of his first 24 shots in the Premier League, Burnley defender James Tarkowski has scored with two of his last three in the competition.
Fulham can maintain momentum at home to Exeter City in the EFL Cup before a weekend trip to Brighton and Hove Albion, while Burnley welcome Olympiacos and Manchester United to Turf Moor in the space of a difficult four days.