Leonardo Jardim's men appeared to be on the way to a second straight defeat five minutes into the second half when defender Mauricio turned into his own net to give Rio Ave the lead.
However, the capital club, who lost 2-0 to arch rivals Benfica in the Lisbon derby in midweek, refused to go away and levelled on 70 minutes as Algeria striker Islam Slimani rose highest to head home Jefferson's pinpoint left-wing cross.
And the visitors completed the turnaround six minutes from time, 19-year-old forward Mane making a crucial impact off the bench.
The teenage forward volleyed home his fourth goal of the campaign to move Sporting into second place, two points behind leaders Benfica and two above Porto ahead of their respective games with Estoril and Vitoria Guimaraes.
Braga also needed late heroics against Arouca as Erik Moreno spared his side's blushes with a 90th-minute goal to earn a 2-2 draw.
Midfielder Custodio had put Braga in front after 50 minutes with his third goal of the season, but the face of the game changed three minutes later as defender Nurio Fortuna was sent off for a second bookable offence.
Arouca duly took advantage and appeared to have snatched the points following a frantic late turnaround.
First substitute forward Roberto levelled matters on 81 minutes, before Spanish attacker Cristian Ceballos netted what looked to be the winner just two minutes from time.
But it was substitute Moreno who would have the final say, the Colombian striking with his first goal since his January move from Millonarios to secure a share of the spoils.
