Froome clocked 51 minutes and 18 seconds for the 37.5km stage from Bourg-Saint-Andeol to La Caverne du Pont-D'Arc, which Dutchman Tom Dumoulin won in 50 minutes and 15 seconds.
Bauke Mollema finished sixth to go second in the overall standings.
There were doubts the stage would go ahead after Thursday's attack in Nice.
Tour officials decided to continue with the race, with a heightened security presence, after at least 84 people were killed when a lorry drove through a crowd celebrating Bastille Day in the southern French city.
There was a sombre atmosphere at both the start and finish lines on stage 13, with most riders not learning of the news from Nice until they woke up on Friday morning.
"We want this day to be a day of dignity as a tribute to the victims," said race director Christian Prudhomme. "The race must continue."
There was a minute's silence held before the first rider set off on the course, and another minute's silence was observed as the jersey holders stood on stage after the day's racing had finished.
The riders, including Froome, placed their bouquets of flowers on an empty podium in a symbolic tribute for the victims in Nice.
Froome, who lives in Monaco, just 12 miles (20km) from Nice, said: "This is definitely a time for people to stand together. It definitely puts things into perspective for us.
"I'm happy with how the stage went but everyone's thoughts are with the people in Nice. It's a special place for me, close to where I'm based, and I can't imagine what everyone is going through."
Stage winner Dumoulin added: "On one side I'm a very happy man but on the other side it's a very sad day.
"I woke up to the terrible news from Nice and there was a question whether we should race. It was a just decision in the end to race. It's terrible what happened and it shadows the day a lot."
On the road, it proved to be a crucial day for 2013 and 2015 champion Froome, as he increased his lead on all of his yellow jersey rivals.
Last year the 31-year-old collected the yellow jersey on stage seven and never relinquished it.
"There will be a strong breakaway group but it will probably come back together and end in a bunch sprint finish"
Read Geraint's preview of stage 14
And since wearing the maillot jaune on stage eight this year, the Team Sky rider has continued to put more time between himself and the other contenders.
His time trial may have been one minute three seconds slower than Dumoulin's superb stage-winning pace, but he was almost two minutes quicker than Mollema, who is now his closest threat for the overall lead.
Fellow Briton Adam Yates, 23, dropped from second to third, but continues to have a remarkable Tour, finishing seven seconds quicker than the highly regarded Nairo Quintana.
Colombian Quintana looks devoid of form, having been unable to chase Froome, Mollema and Richie Porte up Mont Ventoux on Thursday, and now falling to fourth and two minutes 59 seconds off the lead.
The comprehensive stage victory for Dumoulin strengthens the widely held belief that the 25-year-old has the potential to win a future Grand Tour - which includes the Giro d'Italia and La Vuelta.
1.  Tom Dumoulin (Ned/Giant) 50mins 15secs
2.  Chris Froome (GB/Team Sky)+1min 3secs
3.  Nelson Oliveira (Por/Movistar) +1min 31secs
4.  Jerome Coppel (Fra/IAM Cycling) +1min 35secs
5.  Rohan Dennis (Aus/BMC Racing) +1min 41secs
6.  Bauke Mollema (Ned/Trek) +1min 54secs
7.  Geraint Thomas (GB/Team Sky) +2mins
8.  Jon Izagirre (Spa/Movistar) +2mins 2secs
9.  Tony Martin (Ger/Etixx - Quick-Step) +2mins 5secs
10. Steve Cummings (GB/Dimension Data) +2mins 24secs
1.  Chris Froome (GB/Team Sky) 58hrs 2mins 51secs
2.  Bauke Mollema (Ned/Trek) +1min 47secs
3.  Adam Yates (GB/Orica) +2mins 45secs
4.  Nairo Quintana (Col/Movistar) +2mins 59secs
5.  Alejandro Valverde (Spa/Movistar) +3mins 17secs
6.  Tejay van Garderen (US/BMC Racing) +3mins 19secs
7.  Romain Bardet (Fra/AG2R) +4mins 4secs
8.  Richie Porte (Aus/BMC Racing) +4mins 27secs
9.  Daniel Martin (Ire/Etixx - Quick-Step) +5mins 3secs
10. Fabio Aru (Ita/Astana) +5mins 16secs
Britain's Chris Froome extended his lead in the Tour de France to one minute and 47 seconds by finishing second in the stage 13 time trial.