Alcaraz pulled out of the recent Monte Carlo Masters with back and hand problems and struggled at times against the veteran Spaniard as he continues to warm up for the French Open in May.
Top seed Alcaraz breezed past Nuno Borges on Tuesday but found it much tougher going in the last 16 as he continued his Barcelona title defence on the court named after Rafael Nadal.
The 22-time Grand Slam champion is not playing in Barcelona because of a hip injury and announced earlier he will not be fit to take part in the Madrid Masters next week either.
Alcaraz started shakily in windy conditions, saving two break points in the first game, with Bautista-Agut showing plenty of resistance.
The veteran played solidly and competed far better than in his Indian Wells defeat by Alcaraz in March 2022.
Alcarez, looking for his third trophy of the season after triumphing in California this year and again in Buenos Aires, struggled for consistency, making several unforced errors.
The 2022 US Open winner broke for 5-3 and consolidated it to win the first set, producing a spectacular stretching backhand winner in the final game which had fans out of their seats.
Alcaraz broke at the start of the second set and then held his serve for 2-0 and six consecutive games in a row, playing with more confidence.
Bautista dug deep to resist his opponent's pressure and after the pair exchanged breaks in a topsy-turvy second set, took the lead for the first time at 4-5.
However, Alcaraz broke for 6-5 and served out for the victory, clinching progress to the last eight with an ace.
Earlier second seed Stefanos Tsitsipas eased past Denis Shapovalov 6-3, 6-2, to set up a quarter-final clash with Australia's Alex de Minaur.
De Minaur progressed with a walkover after Bulgarian Grigor Dimitrov withdrew due to illness.
Jannik Sinner beat Yoshihito Nishioka 6-1, 4-6, 6-3 and will face Italian compatriot Lorenzo Musetti, who overcame Briton Cameron Norrie 3-6, 6-4, 6-1.
