Appointed on Monday to replace Xabi Alonso, Arbeloa and his side were ousted by Jefte Betancor’s stoppage-time winner at the Carlos Belmonte stadium.
Without French superstar Kylian Mbappe and various other key players, Real struggled against a side currently 17th in Spain’s second division.
Albacete took the lead through Javi Villar but Franco Mastantuono equalised just before half-time.
Jefte put the hosts back ahead after 82 minutes and then scored the winner after Gonzalo Garcia’s 91st-minute goal looked to have forced extra time for the 15-time European champions.
“Here at this club a draw is already bad – it’s a tragedy. Imagine a defeat like this, it’s painful,” Arbeloa told reporters.
“I’m sure all our fans feel the same way. Even more so when it happens against a team from a lower division, although we already know here how tough any opponent can be.
“If anyone is responsible and to blame for this outcome, it’s clearly me, the one who made the decisions regarding the line-up, how we wanted to play, the substitutions.
“I can only thank the players for the way they welcomed me, for the effort they put in today.”
After Real lost the Spanish Super Cup final on Sunday against rivals Barcelona, president Florentino Perez replaced Alonso with reserve team coach Arbeloa.
The Spaniard selected two players currently playing in the reserves he led until Monday – holding midfielder Jorge Cestero and right-back David Jimenez.
As well as Mbappe he left out players including goalkeeper Thibaut Courtois and Jude Bellingham, to rest ahead of the return to La Liga action on Saturday against Levante.
“I’d do the same again, I brought a team capable of winning,” said Arbeloa.
The first half was mostly played under heavy gloom, not just Real’s mood but an intense fog which set in a few minutes into the game.
Vinicius Junior smashed high and wide from long range as neither side produced any clear chances until shortly before half-time when the hosts took a shock lead.
Villar escaped Mastantuono’s attention at a corner and nodded Albacete in front after 42 minutes.
Real levelled before the break through, also capitalising on a corner, with Mastantuono turning home from close range.
Elsewhere, Real Betis beat Elche 2-1 and Alaves defeated Rayo Vallecano 2-0 to reach the quarter-finals.
Holders Barcelona visit second division leaders Racing Santander on Thursday. (AFP)
