Seydou Keita was sent off early on, and the hosts saw Juninho Quixada red carded after 80 minutes, as Barragan, Federico Cartabia and Philippe Senderos secured the triumphValencia made a mockery of being a man down for most of the game to beat Ludogorets 3-0 on Tuesday and stamp one foot in the Europa League quarter-finals.
Barragan's terrific strike five minutes in began what looked to be a routine victory for the visitors, but Seydou Keita's red card as he conceded a penalty threw a potential spanner in the works.
Roman Bezjak stepped up for Ludogrets but los Che shot-stopper Diego Alves leapt to his right to palm away the effort from the spot, then less than 10 minutes later the Bulgarian side were down by two goals when Federico Cartabia turned in a scrappy attack.
The second half was no less entertaining, although the result started to look predictable as Philippe Senderos scored his first goal for the club and Juan Antonio Pizzi's men's third of the night.
Juninho Quixada's sending off with 80 minutes on the clock evaporated any glimmer of hope for a comeback and the hosts will need a miraculous turnaround in the second leg to have any hope of reaching the last eight of the Europa League this season.
Pizzi spent the build-up to the game talking up his side's chances of winning the competition and they got off to a perfect start in Sofia, over 300 kilometres from Ludogorets' usual Razgrad home.
The hosts partially cleared Eduardo Vargas' free kick from the left, but Barragan latched on to the loose ball on the edge of the area and drilled a low half-volley into the bottom-left corner.
Ludogorets were handed a lifeline 24 minutes in as Keita was controversially sent off for clumsily denying Bezjak a goal-scoring opportunity in the area, but Alves - a goalkeeper boasting an excellent record when it comes to saving penalties - kept out the subsequent spot kick.
Valencia narrowly escaped again just after the half-hour mark as Virgil Misidjan hit the post after cutting in from the left, and Pizzi's men capitalised by doubling their lead almost immediately despite Ludogorets' numerical advantage.
Alcacer saw his chipped pass deflect into the area and Cartabia met it with a first-time volley, directing the ball into the bottom-right corner.
Ludogorets finished the half well as Marcelinho and Misidjan both went close from outside the area.
Stoycho Stoev's men also began the second period in determined fashion, putting Valencia under intense pressure, but the Liga side defended well and restricted Ludogorets to long-range efforts before putting the tie beyond all doubt.
Vargas headed a corner from Cartabia back across the face of goal and Senderos was left unmarked to nod the ball into the bottom-right corner.
Ludogorets' day got even worse with 10 minutes to go as Quixada picked up a red card for seemingly kicking out at the theatrical Joao Pereira, allowing Valencia to see out the closing stages with consummate ease.