Angola caused a shock by holding Mexico to a goalless draw, despite playing the last 11 minutes with 10 men.
Mexico hit the post twice and failed to capitalise on the sending off
Mexico enjoyed all the best chances, but a combination of poor finishing and the acrobatics of Angola goalkeeper Joao Ricardo kept them out.
Andre Macanga was sent off on 79 minutes after handling the ball and being shown a second yellow card.
Rafael Marquez then went close with a rasping shot and Omar Bravo hit the post, but Angola managed to hold out.
Mexico, who are ranked fourth in the world and won their opening game against Iran 3-1, went into the match as hot favourites.
And they started confidently, with defender Carlos Salcido setting the tone after just a minute by curling a shot narrowly over the crossbar from 30 yards.
Mexico's cultured captain Marquez then rattled the woodwork with a rasping 30-yard free-kick.
But Angola, prompted by the experienced Figueiredo, became more of a force as the half progressed.
The World Cup debutants had impressed in their opening match, despite losing 1-0 to Portugal, and they again started to play some good football.
After half an hour Figueiredo curled a sumptuous free-kick onto the head of Akwa, but Angola's captain headed wastefully over the crossbar from nine yards out.
Mexico were still impressive in possession, but most of their moves broke down as soon as the ball reached their Argentine-born striker Guillermo Franco.
Coach Ricardo La Volpe responded by bringing on striker Jesus Arellano for Antonio Zinha after 53 minutes.
The substitution paid off almost immediately, when Arellano headed the ball into the path of Franco, whose attempted lob was palmed away by the athletic Joao Ricardo.
Perhaps inevitably considering Portugal's profligacy, it was an Angolan mistake that almost broke the deadlock.
Jamba's clearance hit the back of his own team-mate and the ball rebounded into the path of Bravo, but the Mexican's weak finish was comfortably gathered by Joao Ricardo.
Angola's cause became even more difficult when Andre was shown a second yellow card for handling the ball and became the sixth player to be sent off at this World Cup.
Mexico piled forward in search of the goal that would have put them on course for a place in the last 16.
Marquez hit a rasping 30-yard shot that Joao Ricardo tipped round the bar and then Bravo again spurned a wonderful opportunity by hitting the post from just a yard out.
There were wild celebrations from the Angola fans at the final whistle as their side claimed their first ever World Cup point.
Mexico: Sanchez, Salcido, Marquez, Osorio, Pineda (Morales 78), Torrado, Zinha (Arellano 52), Pardo, Mendez, Franco (Fonseca 74), Bravo.
Subs Not Used: Borgetti, Castro, Corona, Garcia, Guardado, Ochoa, Perez, Rodriguez, Suarez.
Angola: Joao Ricardo, Jamba, Kali, Delgado, Loco, Figueiredo (Rui Marques 72), Macanga, Mateus (Mantorras 68), Mendonca, Ze Kalanga (Miloy 83), Akwa.
Subs Not Used: Airosa, Buengo, Edson, Flavio, Lama, Lebo-Lebo, Love, Marco, Mario.
Sent Off: Macanga (80).
Booked: Delgado, Macanga, Ze Kalanga, Joao Ricardo.
Ref: Shamsul Maidin (Singapore).
Fifa man of the match: Joao Ricardo.
Mexico need a draw against Portugal in the final group match to progress to the last 16 for the fifth successive time in which they have appeared in the World Cup finals. The last occasion they failed to negotiate the group phase was in 1978. Indeed Mexico have only lost three of their last 19 World Cup matches.
Mexico have taken on African nations on 16 occasions winning 10, drawing three and losing three. But at the World Cup, they are still looking for a first win over African opposition, as they lost to Tunisia in 1978, their only previous opponents from the continent.
Angola earned their first ever World Cup point from this goalless stalemate. The draw means that the six World Cup debutants have now recorded two draws and seven defeats. In only three previous World Cup editions was no debutant able to win a single match, and you have to go back to 1982 since no teams making their first appearance in the finals progressed to the second phase. When Angola midfielder Andre Macanga received the second yellow card in this game, it marked the 1,500th caution in World Cup history.