Organisers had debated whether the stage should be cancelled and officials observed a minute's silence before the first rider went down the ramp. There was another minute of silence at the podium ceremony after the stage.
Dutchman Dumoulin (Giant-Alpecin) clocked 50 minutes 15 seconds, beating Froome by 1:03.
The Briton, however, extended his overall race lead to 1:47 over Dutchman Bauke Mollema (Trek-Segafredo), who moved up to second but lost 51 seconds to the Team Sky rider.
Froome's compatriot Adam Yates (Orica-Bike Exchange) dropped from second to third, 2:45 off the pace after battling strong head and crosswinds.
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Colombian Nairo Quintana (Movistar) lost 2:04 on the day to Froome, to slip from third to fourth overall, 2:59 off the pace ahead of a gruelling final week in the Alps.
The 13th stage of the signature event of the country's sporting summer began at 0905 London time, with its publicity caravan -- usually playing loud music to energise the thousands lining the route -- silenced for the day.
A gunman at the wheel of a heavy truck ploughed into a crowd celebrating France's national day in the southern city late on Thursday, killing at least 84 people and injuring scores in what President Francois Hollande called a terrorist act.
"I woke up this morning and learnt something terrible had happened. It’s an extremely sad day. So it was a stage with two sides," said Dumoulin following his second stage win of the Tour after he prevailed in a mountain-top finish at Arcalis, Andorra last Sunday.
"There was a question this morning whether we could start. And I think it was right to go on racing. It’s horrific what happened but we cannot let terrorists decide on how we should be living.
"Two hours before my start it was decided to race and I decided to do everything I could to race well because that’s the only thing I can do. I did a very good TT (time trial). I did a good time. I cannot say anything about the others. It was a very good TT for me personally.”
Frenchman Thibaut Pinot, who was third in the 2014 Tour de France, but dropped out of contention in the first mountain stage this year, abandoned the race on Friday because of illness.
Froome lacked the juice to beat time trial specialist Dumoulin but showed no ill effects of the crash he suffered in the finale to Thursday's 12th stage, when he retained the yellow jersey thanks to a jury decision that cancelled the time he had lost in the incident.
(Reporting by Julien Pretot; Editing by Toby Davis)
