The Swedish international striker was one of Mourinho's key summer signings after taking charge at Old Trafford and after scoring the winner in last weekend's FA Community Shield clash against Leicester grabbed the third in the 3-1 win at the Vitality Stadium.
Ibrahimovic has previously scored on league debuts for Juventus, Inter Milan, Barcelona and Paris St-Germain, striking home from just outside the penalty area with a low shot.
United was already 2-0 ahead after goals either side of half time from Juan Mata and Wayne Rooney. Adam Smith hit a late consolation for the home side, who woke up too late in its EPL opener.
Another acquisition, Henrikh Mkhitaryan from Borussia Dortmund, also made his debut as a second half replacement for Mata, but record signing Paul Pogba had to sit it out after picking up two bookings for previous club Juventus.
Mata, rumored to be surplus to requirements at United, was preferred to start by Mourinho and justified his inclusion by taking advantage of a double blunder from home defender Simon Francis.
First his backpass saw Mata reach the ball ahead of home goalkeeper Artur Boruc, then the rebound hit Francis to leave the Spanish midfielder with the simplest of chances to score on 40 minutes.
It was the first significant action of a scrappy half and the standard did not improve greatly after the break, with Rooney putting the visitors further ahead just before the hour.
It was hardly a classic, Antonio Valencia crossing for Anthony Martial to scuff his shot that fortunately bounced up for Rooney to steer a clever header past Boruc.
Ibrahimovic's sweetly struck effort soon afterwards injected a bit of quality, but the goal of the game came from Smith after fine work from Lewis Grabban.
David de Gea was called into late action in the United goal as Bournemouth started to show the type of form that helped them survive its first season in the EPL, but the three points was never in doubt.
Mourinho said after the match that "everything" about the United performance pleased him but would also have been delighted by the solid debut at the back of another new boy Eric Bailly from Villarreal.
Later Sunday saw a seven-goal thriller at the Emirates where Liverpool beat home side Arsenal 4-3.
Arsenal took the lead through Theo Walcott, who had seen an earlier penalty saved before Jurgen Klopp's visitors hit back in devastating style to lead 4-1.
Brazilian Philippe Coutinho was instrumental in the comeback, equalizing with a trademark free kick just before the interval.
Coutinho was involved as Adam Lallana put put Liverpool ahead before Coutinho added his second and a third from close range after a 19-pass movement.
Summer signing Sadio Mane appeared to have made the three points safe with a surging run and shot for 4-1 until Arsene Wenger's side made a belated comeback.
Substitute Alex Oxlade-Chamberlain scored in similar style to Mane, but with Simon Mignolet at fault, before Calum Chambers headed in a third.
It added some gloss on a home display which had home fans chanting "you don't know what you are doing" in the direction of Wenger, reminiscent of last season's opening day defeat.
The opening day of the season saw a shock defeat for last season's surprise champions Leicester City.
Visiting promoted and managerless Hull City, Leicester was beaten 2-1, the first time a reigning champion had been beaten in a season opener since Arsenal at Manchester United in 1989.
Adama Diomande put Hull ahead before Riyad Mahrez leveled from the penalty spot at the start of the second half after Demarai Gray had been fouled.
Hull grabbed a vital three points when Robert Snodgrass drilled home his shot after after a poor clearance.