Raphael Lagarde's late penalty looked to have won it for the home side, but Louw crossed the line in the 79th minute to put Bath back in front and Stephen Donald's conversion added the gloss to the scoreline.
The visitors fell behind in the third minute when Benjamin Petre scored a try for Agen and the accurate Lagarde converted.
Donald and Lagarde exchanged penalties as Agen extended their lead to 10-3, but Michael Claassens scored a crucial try for Bath just before half-time to make it 10-8 to the French side at the interval.
Another Lagarde penalty extended Agen's lead at the start of the second period before Kyle Eastmond crossed for Bath's second try to level the score and Donald's conversion nudged Bath in front for the first time.
Lagarde restored Agen's advantage and, although Donald crossed for Bath in the 54th minute, he missed the conversion and it threatened to cost Bath dear when Lagarde pulled Agen back within a point.
Lagarde's 71st-minute penalty gave the hosts a 22-20 advantage which looked like being enough until Louw's late strike maintained Bath's 100 per cent start to the group.
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