The nine times world rally champion moved up to second place overall from fourth, six minutes and 55 seconds behind, with Spaniard Carlos Sainz third in a one-two-three for the French manufacturer.
Qatar's Nasser al-Attiyah, a two times Dakar winner and leader after the opening day, fell to fourth in his Toyota and was 58 minutes adrift of 13 times winner Peterhansel, who was third in the stage.
"It was a very difficult day. We had two flat tyres at the beginning and afterwards we took care because we didn’t have any spares," al-Attiyah told the Dakar website.
"At the first dunes, we went into a big hole and got stuck for 30 minutes, before it happened a second time. We lost pressure in one of the tyres. We tried to put air back in and we were stuck again for 30 or 20 minutes."
Peterhansel also suffered low tyre pressure and had to stop and make a change, losing several minutes. He then made a navigation error towards the end.
Cyril Despres, who had led after the second stage, saw his chances of winning a sixth title after five on bikes effectively disappear when he broke a rear wheel on his Peugeot and had to wait for assistance.
In the bikes, Sunderland crashed while leading and had to be flown out by helicopter.
His exit meant that Yamaha's French rider Adrian van Beveren took over the lead, one minute and 55 seconds clear of Chilean Pablo Quintanilla, after winning the stage.
Wednesday's fifth stage sees the bikes and quads take a different route to the cars and trucks, with all ending up in Arequipa on the last full day in Peru before heading for Bolivia.
(Reporting by Alan Baldwin, editing by Ken Ferris)
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