Casemiro's (31) spectacular first-half goal proved decisive on the south coast with United now needing just one point from their two remaining Premier League games this season - both at Old Trafford -- against London duo Chelsea and Fulham next week to be assured of a place among European club football's elite.
The three-time European champions would have booked their spot on Saturday had their win been accompanied by a defeat for Liverpool.
But United's arch-rivals came from behind to secure a 1-1 draw at home to Aston Villa.
Not that United manager Ten Hag was that concerned ahead of Thursday's encounter with a struggling Chelsea side.
"It (Liverpool's result) is not important, I'm telling you already for weeks it's not about our opponents," the Dutchman insisted.
"We are in the lead, it's only up to us, we have everything in our hands, we don't have to look behind. We have to make our performance."
Qualifying for the Champions League would mean Ten Hag had achieved a key target in his first season in charge, with United having already won the League Cup ahead of next month's FA Cup final against local rivals Manchester City.
Marcus Rashford (25) missed out for three-times European champions United with an unspecified illness, the England forward having also been sidelined from last week's 2-0 win over Wolves due to a leg injury.
But despite his absence, United took just nine minutes to go ahead on the south coast.
'Brilliant Casemiro'
Christian Eriksen's (31) floated cross into the box was flicked into the path of Casemiro by defender Marcos Senesi (26) before the Brazilian midfielder hooked in an acrobatic volley.
"He keeps surprising us, Casemiro," said Ten Hag.
"He's such a brilliant football player. We know him as very good (at) organisation, positioning, anticipating, intercepting a lot of balls, winning duels, a fighter.
"Also in possession, his passing and distribution... We're happy we found him and his contribution is massive."
Dominic Solanke (25), however, could have equalised for Bournemouth but his low drive was well-saved by David de Gea (32), whose 16th clean sheet of a sometimes difficult season ensured he won the Premier League's Golden Glove goalkeeping award.
David Brooks (25), making his first start since being diagnosed with cancer in 2021, almost levelled after the interval only for De Gea to turn his fierce drive over the bar.
The Cherries also missed two late chances to equalise, substitute Kieffer Moore (30) shooting straight at De Gea, with only the keeper to beat, before Senesi volleyed onto the roof of the net in added time.
Bournemouth boss Gary O'Neil, however, was proud of how his side had secured their top-flight status before kick-off.
"I don't think anyone could have complained if Kieffer Moore had scored and it had finished 1-1," he said.
"I think everyone felt the Everton game (next week) would be pivotal. The fact that it isn't shows what a big achievement it's been from the players to stay up with four games to spare."
