South Africa coach Shakes Mashaba had urged his team to secure the three points they needed on Saturday to ensure they would feature in next year's tournament in memory of murdered captain Senzo Meyiwa.
Goalkeeper Meyiwa was tragically shot dead last month in an altercation involving two attackers on the outskirts of Johannesburg last month.
There was a minute's silence before kick-off at Moses Mabhida Stadium in the first game Bafana Bafana have played since Meyiwa's passing and goals from Serero and Tokelo Rantie ensured Mashaba's side will feature in the showpiece which will now be held in Equatorial Guinea next year.
Salah Ibrahim pulled a goal back for Sudan to set up a tense final, but South Africa, who were captained by Dean Furman, held on to stay top of Group A with one game against Nigeria remaining and are guaranteed a top-two finish.
Sudan, on the other hand, have no chance of qualifying for the tournament having secured only three points from their five matches but will feel they should have been awarded a penalty in time added on
Backed by a partisan crowd in a game which was moved from Nelspruit to Meyiwa's home city at the request of Mashaba, the hosts showed positive intent in the opening stages but were indebted to a superb last-ditch challenge from Sibusiso Khumalo to thwart Mudather El Tahir as he raced clear.
Ajax midfielder Serero eased the tension seven minutes before the break, putting the hosts in front with a right-footed finish to round off a lovely passing move.
Sudan striker Bakri Al Madina was fortunate to get away with catching South Africa goalkeeper Darren Keet with a late high boot, then Bongani Ndulula was guilty of a glaring miss for the hosts with the goal gaping.
Visiting keeper Moez Mahjoub was quickly off his line to make a brave save to deny Rantie just after the break as Mashaba's men attempted to give themselves some breathing space.
Rantie was not to be denied after 53 minutes, though, as he picked up a pass and skipped pass two players before applying a clinical right-footed finish.
Ibrahim sent home nerves jangling when he volleyed home from inside the penalty area, and Sudan were denied a late penalty as Khumalo felled his man inside the area but South Africa held on to spark celebrations on and off the pitch.
