Davis nudged Bath ahead in the ninth minute but an impressive Ulster scrum yielded a quick levelling penalty by Humphreys.
A superb dart by Ulster full-back Clinton Schifcofske then set up Ulster's first try in the 14th minute with Nagusa diving over a ruck on the edge of the Bath line to score like a back-row forward.
Humphreys added a superb conversion from the touchline to extend the advantage to 10-3.
Davis's second penalty reduced Ulster's lead to 10-6 at the break as the home side briefly lost their momentum by conceding a few needless penalties.
Ulster started the second half strongly and a series of pick-and-drives instigated after a line-out take by Stephen Ferris resulted in Humphreys calmly slotting a drop-goal.
Davis reduced Ulster's lead to four points by notching another penalty but the contest was effectively over when Faloon repeated his try-scoring heroics of last week's Magners League win over Scarlets.
Humphreys' superb offload set up the attack and hooker Andy Kyriacou drew the defenders before releasing Faloon to score under the posts.
Davis responded almost immediately with a Bath penalty but a comeback never looked on and Humphreys added two more Ulster penalties before the finish.
The last score came after Bath's Andy Beattie had been red carded for what appeared to be dangerous use of the boot.
Ulster coach Brian McLaughlin:
"I'm really happy with the way things went and how we worked so hard for each other.
"I was very, very pleased with our match management which was superb.
"I think this squad of players are hopefully building towards something, but we still have a long way to go."
Bath coach Steve Meehan:
"It's hard to complain about it as I haven't seen it yet but it's really out of character for Andy Beattie and if there was an action it came probably more out of frustration.
"I felt our first half performance was very good and the guys were confident in the changing room, but unfortunately for us we made one or two errors which put us on the back foot.
"As a result, Ulster fed off our errors and played well. We just had to deal with too much pressure."
Ulster: C Schifcofske, T Nagusa, D Cave, P Wallace, A Trimble, I Humphreys, I Boss, T Court, A Kyriacou, B Botha, E O'Donoghue, R Caldwell, S Ferris, W Faloon, R Diack. Replacements: Danielli for Nagusa (73), Young for Court (55), Brady for Kyriacou (73), D Fitzpatrick for Botha (66), Henry for Diack (61).
Not Used: McComb, Marshall, Whitten.
Bath: N Abendanon, J Maddock, M Carraro, S Hape, M Banahan, R Davis, M Claassens (c), D Flatman, L Mears, D Bell, S Hooper, D Grewcock, A Beattie, J Salvi, J Faamatuainu. Replacements: Cuthbert for Maddock (71), Bemand for Claassens (73), Barnes for Flatman (67), Dixon for Mears (41), Skirving for Faamatuainu (67).
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