West Ham striker Sakho opened the scoring off a pass from Liverpool attacker Mane and both were involved in the move that led to a Thamsanqa Mkhize own-goal before half-time.
Victory gave Senegal an uncatchable five-point lead over Burkina Faso and Cape Verde in Group D with one round left.
The only previous World Cup appearance by Senegal was in 2002 when current coach Aliou Cisse captained a team that reached the quarter-finals in South Korea.
Bottom-of-the-table South Africa lie seven ponts behind Senegal ahead of a return match in Dakar next Tuesday, with Burkina Faso hosting Cape Verde at the same time.
South Africa defeated Senegal 2-1 in Polokwane last November, but FIFA ordered a replay because of "match manipulation" by Ghanaian referee Joseph Lamptey, who was banned for life.
Senegal are the third African qualifiers for the World Cup in Russia after Nigeria and Egypt, who secured places during the penultimate qualifying round last month.
The other two slots will be filled Saturday with north African countries Tunisia and Morocco needing draws to qualify.
Tunisia host out-of-contention Libya in Group A holding a three-point advantage over the Democratic Republic of Congo, who play Guinea in Kinshasa.
Morocco are one point in front of the Ivory Coast ahead of a showdown in Abidjan with the visitors hoping to end a 20-year absence from the World Cup.
South Africa goalkeeper and captain Itumeleng Khune played with a protective white mask after fracturing a facial bone in a club match last weekend.
Senegal took the lead after 12 minutes when Sakho raced on to a Mane pass and side-footed the ball to the right of onrushing Khune.
Stung by falling behind so early, South Africa sought an equaliser and came close midway through the opening half at Peter Mokaba Stadium.
Goalkeeper Khadim Ndiaye failed to grasp a fierce close-range Themba Zwane shot, but recovered to catch the rebound after the ball struck inrushing Percy Tau.
South Africa came even close soon after when Lebogang Manyama swept a low cross against the crossbar and Senegal hastily cleared the loose ball.
The Senegalese Teranga Lions doubled their lead after 38 minutes against the run of play with another goal involving Sakho and Mane.
Sakho broke clear and crossed to Mane, whose shot was blocked by Khune only for the clearance to hit Mkhize and trickled back into the net.
As half-time approached, Manyama went close again to scoring, this time from a free-kick that Ndiaye tipped over.
South Africa introduced midfielders Andile Jali and Siphiwe Tshabalala midway through the second half, but neither made an impact.