Team Sky's Thomas came home 21 seconds behind Spain's Bilbao, who staged a heroic solo effort 12 kilometres from the finish to triumph for Astana.
Thomas finished fifth on the final day in the end, having moved into the yellow jersey and clear of his rivals thanks to second place finishes at Valmorel and La Rosiere on stages five and six.
Briton Adam Yates, riding for Mitchelton, is second overall, 1min 29sec behind Thomas. Now I realise I've won a massive race.
"I can rest a little bit now and wait for July". Bilbao made his move late on during the gruelling stage that is nearly the same as stage 11 in the Tour de France next month, with four major climbs and a summit finish at la Rosiere. "I managed to get a few more bonus seconds which is good".
"My role in the team will be as one of the protected guys, Froome is going to be one of the leaders".
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"It's been two back-to-back hard stages", he added. France's Romain Bardet, of AG2R, jumped to third from seventh overnight and is at 2min 01sec.
Adam Yates was extremely happy to win the stage after a hard week of racing.
The 32-year-old Thomas registered his first win at the Dauphine, probably the greatest success of his professional career, topping even his Paris-Nice victory in 2016.
He said it was learning from the previous days that helped him take victory on St Gervais Mont Blanc, delaying his attack to claim the stage win.
Thomas now leads Yates by 1:29 in the overall standings.