Visitor Lyon is second in the table, with Agen second from bottom, but failed to score in the first half, trailing 12-0 at the break.
Jonathan Pelissie was sent to the sin bin as he conceded an early penalty try, before Andres Zafra Tarazona crossed, and Lyon ended the half with Liam Gill having also seen yellow.
The high-flying away team was still behind heading into the closing stages, despite a pair of Jonathan Wisniewski penalties.
But Chiocci went over in the 75th minute, and Wisniewski kept his nerve with the conversion to seal a tense 13-12 victory.
Lyon still trails Bordeaux Begles after the Top 14 leader brushed aside Bayonne 22-3.
Bordeaux was in control from the first minute when Jean-Baptiste Dubie scored, and further tries came from Kane Douglas and Remi Lamerat before a scoreless second half.
Jules Plisson had a try, four conversions and three penalties for third-placed La Rochelle in a 44-13 success at Pau.
Racing 92 is back in fourth and beat Clermont Auvergne 27-19, as Teddy Iribaren similarly starred with a try, three conversions and a pair of penalties.
Elsewhere, Thomas Darmon's 75th-minute try meant a late Brive response mattered little, Montpellier winning 29-26.