Chris Boyd's side secured a stunning 43-39 comeback success against the Dragons at Rodney Parade on Saturday.
And now they will host an Ulster outfit buoyed by a huge last-16 win at Harlequins.
The Pro 14 team thrashed Quins 57-21 on Sunday evening, scoring eight tries to set up the meeting with Saints.
Ulster finished second in Pro 14 Conference A, seven points behind Leinster.
Challenge Cup quarter-finals
Saints v Ulster
Bath v London Irish
Leicester Tigers v Newcastle
Montpellier v Benetton