Tom Dumoulin (Giant-Alpecin) took his second stage win of the 2016 race, recording a time that was head and shoulders above everyone else.
Froome gained two minutes and five seconds on Nairo Quintana (Movistar), who put in a below-par performance to finish 20th on the stage and is now in fourth overall.
After Froome, Bauke Mollema (Trek-Segafredo) was the best performing of the general classification men, placing fifth on the stage.
Mollema moves up to second overall, 1-47 behind Froome, as British rider Adam Yates (Orica-BikeExchange) slips to third at 2-45. Yates put in a solid ride to finish 18th on the stage.
Romain Bardet (Ag2r), Fabio Aru (Astana) and Joaquim Rodriguez (Katusha) were the biggest time losers of the day, all slipping down the overall classification.
Geraint Thomas (Sky) benefitted from a good run to claw back some time and move up to 15th overall, 6-48 behind Froome. Australian Richie Porte (BMC) rises to eighth overall.
Dumoulin's arrival at the finish line saw him smash the previous best time set by Nelson Oliveira (Movistar) by over a minute and a half, raising the bar significantly. Oliveira held on for third behind Froome.
A combination of fatigue from the previous two weeks' effort and a hilly course saw several of the traditional time trial experts fail to register quick times at the finish. Fabian Cancellara (Trek-Segafredo), Tony Martin (Etixx-QuickStep) and current TT world champion Vasil Kiryienka (Sky) were well off the pace.
Watch: Tour de France stage 13 highlights
The podium celebrations were truncated with a minute's silence observed as a mark of respect for those that lost their life in the previous evening's tragedy in Nice.
Simon Gerrans (Orica-BikeExchange) and French TT champion Thibaut Pinot (FDJ) were absent from the list of 188 starters, having withdrawn overnight. Gerrans fractured his collarbone in a crash the previous day, and Pinot was reportedly suffering from bronchitis. Edward Theuns (Trek-Segafredo) abandoned during the stage after crashing.
The 2016 Tour de France continues on Saturday with stage 14 on a route that heads due north from Montélimar to Villars-les-Dombes Parc des Oiseaux. It's a relatively flat route, and on papre looks to be one for the sprinters. However, the high winds may play a part in proceedings just as they have affected the race in the past few days.
Tour de France 2016, stage 13: Bourg-Saint-Andéol to La Caverne du Pont-d'Arc, 37.5km ITT
1. Tom Dumoulin (Ned) Giant-Alpecin in 50-15.14
2. Chris Froome (GBr) Team Sky at 1-03
3. Nelson Oliveria (Por) Movistar at 1-31
4. Jerome Coppel (Fra) IAM Cycling at 1-35
5. Rohan Dennis (Aus) BMC Racing at 1-41
6. Bauke Mollema (Ned) Trek-Segafredo at 1-54
7. Geraint Thomas (GBr) Team Sky at 2-00
8. Jon Izaguirre (Esp) Movistar at 2-02
9. Tony Martin (Ger) Etixx-QuickStep at 2-05
10. Steve Cummings (GBr) Dimension Data at 2-14
15. Alejandro Valverde (Esp) Movistar at 2-48
16. Tejay van Garderen (USA) BMC Racing at 2-50
18. Adam Yates (GBr) Orica-BikeExchange at 3-01
20. Nairo Quintana (Col) Movistar at 3-08
21. Richie Porte (Aus) BMC Racing at 3-08
23. Fabian Cancellara (Sui) Trek-Segafredo at 3-15
30. Romain Bardet (Fra) Ag2r at 3-52
33. Daniel Martin (Irl) Etixx-QuickStep at 4-10
37. Fabio Aru (Ita) Astana at 4-25
42. Joaquim Rodriguez (Esp) Katusha at 4-46
General classification after stage 13
1. Chris Froome (GBr) Team Sky in 58-02-51
2. Bauke Mollema (Ned) Trek-Segafredo at 1-47
3. Adam Yates (GBr) Orica-BikeExchange at 2-45
4. Nairo Quintana (Col) Movistar at 2-59
5. Alejandro Valverde (Esp) Movistar at 3-17
6. Tejay van Garderen (USA) BMC Racing at 3-19
7. Romain Bardet (Fra) Ag2r at 4-04
8. Richie Porte (Aus) BMC Racing at 4-27
9. Daniel Martin (Irl) Etixx-QuickStep at 5-03
10. Fabio Aru (Ita) Astana at 5-16
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