The south coast side have never featured in the last eight and will now host eight-time winners Liverpool at the Goldsands Stadium.
Derby have a home tie against Premier League leaders Chelsea, while League One Sheffield United welcome high-flying Southampton.
Newcastle, who knocked out holders Manchester City, travel to Tottenham.
The ties will be played in the week beginning 15 December.
Sheffield United reached the semi-finals of the FA Cup last season and are enjoying another good cup run after knocking out MK Dons in the previous round.
Championship leaders Derby are rewarded for coming from 2-0 down to beat Fulham 5-2 with a home draw against the Premier League leaders.
Bournemouth - the third of the non-Premier League sides left in the competition - eliminated West Brom on Tuesday.
Full quarter-final draw:
Derby v Chelsea
Tottenham v Newcastle
Bournemouth v Liverpool
Sheffield United v Southampton
Championship side Bournemouth will play Liverpool in the quarter-finals of the Capital One Cup.