The pair came in after November's sacking of Markus Schopp, but had a break clause in their contracts which could be exercised if the club were relegated.
The Reds have decided to do that after their relegation from the Championship was confirmed on Friday night.
Asbaghi left his job as coach of Sweden's Under-21s to take over at Oakwell with the hosts four points adrift of safety after 17 matches. They were 14 points above a Derby County side whose points tally was still in the minuses after punishments over financial issues at Pride Park.
The 36-year-old won six and lost 17 of his 28 matches and left with the club bottom of the table, and relegation confirmed after a 2-1 defeat at Huddersfield Town.
Just five of the points he gleaned came away from home, with the only win at Hull City. Another victory came against League Two Barrow in the FA Cup, but only 5-4 after extra-time.
It was a huge comedown for a club who reached the play-off semi-finals last season but the tone for 2020-21 was set before Asbaghi's arrival.
Valerien Ismael left for West Bromwich Albion and took key members of his coaching staff and captain Alex Mowatt with him. Striker Daryl Dike, who had been on loan from Major League Soccer, did not return. At the same time, chief executive Dane Murphy left.
Barnsley's recruitment of replacements was very poor. Schopp tried to change from Ismael's more direct style of football with no success, and big summer signings Obbi Oulare and Aaron Leya Iseka arrived unfit. Leya was unable to have a significant impact when he got fit, Oulare never really did, and was loaned out in January.
Asbaghi improved things a little, but not enough to spare him from criticism from the terraces.
The January deadline-day loan signings of Domingos Quina and Amine Bassi gave the team more attacking threat but it was too little, too late.
Sibila took charge of the team for a short spell in January when Asbaghi was isolating with Covid-19.
Naturally Asbaghi was asked about his future after the game, but said he would be taking Monday's press conference for the Tuesday Championship game at home to Blackpool, and that the club would have to spend the remainder of the season thinking hard about its future and identifying what had gone wrong.
They appear to have reached their first conclusion quickly.
Despite Asbaghi's Friday-night comments, the decision has been described as being reached "by mutual consent".
Asbaghi described his time at Oakwell as "a real privilege.”
Chief executive Khaled El-Ahmad stressed: “Our search for a new coach has already started and an announcement will follow once that process is concluded.”
Martin Devaney will take charge for the rest of the season, assisted by Tom Harban and Jo Laumann.