Mina - who scored in the 3-0 win over Poland - headed in with 16 minutes left on the clock to seal Colombia the points they needed to top the group.
Senegal were denied a first-half penalty that referee Milorad Mazic initially awarded, but overturned after a VAR review.
Mazic pointed to the penalty spot after defender Davinson Sanchez tackled Senegal's Sadio Mane near the penalty spot in the 16th minute.
However, he overturned the call after reviewing video, determining that Sanchez's heel touched the ball away before he made contact with Mane.
With the game goalless as Colombia's World Cup dream appeared to be slipping away, Mina got another World Cup goal to spark scenes of jubilation among his nation's fans to seal a spot in the last 16.
Colombia's success was all the more remarkable for the fact that they had to do it without James Rodriguez after the star midfielder was substituted off just 31 minutes into the game.
James' team-mate Radamel Falcao patted him on the back as he walked slowly off the pitch with his head bowed and headed straight for the locker room, with Luis Muriel replaced him.
James has been dealing with a nagging calf injury and soreness that kept him from starting Colombia's tournament-opening loss to Japan.