Iraola was given a warm reception at Bournemouth after leading the Cherries into Europe for the first time in the club's history.
Liverpool win at Bournemouth on Iraola's return
Alexander Isak's second half goal proved the difference as Liverpool secured three points on the South Coast.
That success led to the Basque being snapped up by Liverpool but he has endured an up and down start at Anfield and needed a victory to ease tensions before a three-week international break.
Liverpool's performance was again far from convincing but they got the only goal when Cody Gakpo's cross sat up for Alexander Isak to fire in his fourth of the season.
Victory moves Liverpool up to sixth and within a point of third-placed Brighton.
