Belgian Teuns and Wanty-Gobert's Martin of France ended 13 seconds ahead of the peloton despite a late attempt to rein them in by the chasing pack which included Britain's Chris Froome, who is looking for a record fourth overall win in the eight-stage race.
Teuns, 27, claimed the yellow jersey from Norway's Edvald Boasson Hagen who won the opening stage on Sunday but could only cross the finish line 1min 48sec behind on the second.
Four-time Tour de France winner Froome is preparing for this year's Grand Boucle and is a part of six-man group 24sec behind Teuns on the general leaderboard.
Tuesday's 177km third stage heads north through rolling hills from Le Puy-en-Velay to the town of Riom outside Clermont-Ferrand.
Results after Monday's second stage of the Criterium du Dauphine over 180km from Mauriac to Craponne-sur-Arzon:
1. Dylan Teuns (BEL/BAH) 4hr 12min 41sec, 2. Guillaume Martin (FRA/WGG) same time, 3. Jakob Fuglsang (DEN/AST) at 13sec, 4. Thibaut Pinot (FRA/FDJ) at same time, 5. Michael Woods (CAN/EF1) s.t., 6. Alexey Lutsenko (KAZ/AST) s.t., 7. Petr Vakov (CZE/DEC) s.t., 8. Nairo Quintana (COL/MOV) s.t., 9. Wout Poels (NED/INE) s.t., 10. Adam Yates (GBR/MIT) s.t.
11. Chris Froome (GBR/INE) at 13sec; 29. Edvald Boasson Hagen (NOR/DDT) at 1min 49sec
1. Dylan Teuns (BEL/BAH) 7hr 37min 03sec, 2. Guillaume Martin (FRA/WGG) at 03sec, 3. Jakob Fuglsang (DEN/AST) 20, 4. Alexey Lutsenko (KAZ/AST) 21, 5. Nairo Quintana (COL/MOV) 24, 6. Michael Woods (CAN/EF1) same time, 7. Wout Poels (NED/INE) s.t., 8. Chris Froome (GBR/INE) s.t., 9. Thibaut Pinot (FRA/FDJ) s.t., 10. Adam Yates (GBR/MIT) s.t.