Italian Conti rode away from a 12-strong breakaway in the final 20km to win the longest stage of the race - 213.4km from Bilbao to Dantxarinea.
Froome and Colombian Quintana finished in the peloton almost 34 minutes later.
They were happy with a more sedate day in the saddle because both weekend stages feature mountain-top finishes.
Saturday's 196km stage includes four categorised climbs in the Pyrenees and culminates with a 21km ascent of the Col d'Aubisque, while Sunday's 118km stage finishes with a 14.5km ascent to Formigal.
Team Sky's Froome is aiming to become the first rider in 38 years to win the Tour de France and Vuelta in the same year.
Movistar's Quintana, who finished third behind Froome at the Tour, has won one Grand Tour, the 2014 Giro d'Italia.
Lampre-Merida rider Conti finished 55 seconds ahead of Switzerland's Danilo Wyss, who won the five-man sprint for second place.
"It's my first victory in a Grand Tour and it's a great feeling," said 23-year-old Conti.
Stage 13 result:
1. Valerio Conti (Ita/Lampre)               5hrs 29mins 04secs
2. Danilo Wyss (Swi/BMC Racing)       +55secs
3. Sergey Lagutin (Rus/Katusha)                     same time
4. Vegard Laengen (Nor/IAM Cycling)
5. Michael Gogl (Aus/Tinkoff)
6. Yves Lampaert (Bel/Etixx - Quick-Step)
7. Cesare Benedetti (Ita/BORA)              +1min 02secs
8. Jelle Wallays (Bel/Lotto)              +1min 04secs
9. Gatis Smukulis (Lat/Astana)                      same time
10. Stephane Rossetto (Fra/Cofidis)         +1min 08secs
Selected:
13. Alejandro Valverde (Spa/Movistar)        +33mins 54secs
17. Nairo Quintana (Col/Movistar)                                     same time
21. Peter Kennaugh (GB/Team Sky)
28. Chris Froome (GB/Team Sky)
General classification after stage 13:
1. Nairo Quintana (Col/Movistar)          52hrs 56mins 29secs
2. Chris Froome (GB/Team Sky)             +54secs
3. Alejandro Valverde (Spa/Movistar)         +1min 05secs
4. Esteban Chaves (Col/Orica)             +2mins 34secs
5. Alberto Contador (Spa/Tinkoff)            +3mins 08secs
Selected:
7. Simon Yates (GB/Orica)                 +3mins 25secs
15. Peter Kennaugh (GB/Team Sky)           +6mins 30secs
Britain's Chris Froome continues to trail leader Nairo Quintana by 54 seconds after Valerio Conti broke clear to win stage 13 of the Vuelta a Espana.