America had already qualified as Group Two winners. The top two in each of the three groups go through along with the two next best-placed teams, one of which was Pumas in third spot in Group Three.
Veron, who netted a loose ball in the 12th minute, said: "Every time I play against America I get to score.
"I think the team made a great effort to get these three points that got us our qualification," he told Televisa.
The victory was bad news for Tigres UANL, who had only managed to draw 2-2 away to Atlas, the team with the worst record, on Saturday, but would have gone through if Pumas had not won.
Title holders Toluca, who won the Clausura championship in the second half of last season, failed to make the cut, finishing fourth in Group Two after a 1-1 draw with Atlante.
Their coach Jose Manuel de la Torre, who is now moving to his new job as Mexico coach, said the championship had been a failure for the team he steered to the Clausura title.
"For a team coming from being champions to not qualifying in the next tournament, of course it's a failure," De la Torre told reporters.
The other third-placed qualifiers were Concacaf Champions League title holders Pachuca from Group Two, who will go on to play at the Club World Cup in Abu Dhabi next month.
Pumas will meet favourites Cruz Azul, who finished top of the overall standings, in the quarter-finals with the first leg scheduled for Wednesday.
The other line-ups are: Monterrey v Pachuca, Santos Laguna v Chiapas and America v San Luis.
Guadalajara, who rival America for popularity and success, failed to qualify despite a 3-2 away win over Group One winners Monterrey on Saturday.
