Goals after the break by Germany's Julian Draxler and a strange second from Cameroon defender Joel Matip put Schalke in the last 16 to ease the pressure on under-fire coach Jens Keller.
The 43-year-old has been told a decision on his future will be made over the winter break in the wake of the Royal Blues' erratic form this season.
Having been knocked out of the German Cup last week and trailing leaders Bayern Munich by 17 points in the Bundesliga, Keller would have been set for dismissal after a year in charge had Schalke failed to go through.
The Royal Blues faced ten men for just under an hour as FC Basel had centre-back Ivan Ivanov, who cleared an early shot off the line, sent off for holding back Schalke's Adam Szalai on 31 minutes.
The Hungarian striker had only just replaced Schalke captain Benedikt Hoewedes, who went off with a hamstring injury just approaching the half-hour mark.
Hoewedes was lucky not to be shown a red card, instead of the yellow he received on 29 minutes, as he injured himself with a rugby-style tackle on Basel captain Marco Streller, who would have been clear on goal.
Schalke had the best chance of the first half when Jefferson Farfan's cross was cleared onto the crossbar by Basel defender Taulan Xhaka on 12 minutes.
The ball fell to Julian Draxler who squared it, but 18-year-old Max Meyer's shot was cleared off the line by Ivanov.
The breakthrough Schalke desperately needed came on 50 minutes when Farfan swung in a cross and found Draxler unmarked at the far post to slam home his shot and give the Royal Blues the lead.
Schalke secured their passage to the knock-out phase seven minutes later when their second goal arrived in bizarre fashion.
Basel sprang their offside trap, but it failed to prompt a flag from the linesmen.
Replays showed with five Schalke players offside, Matip found himself alone in the penalty area from Farfan's free-kick and he powered his shot past Basel goalkeeper Yann Sommer.
Basel coach Murat Yakin was left protesting incredulously on the sidelines, but the defeat means the Swiss champions finish third in the group to drop into the Europa League.
With champions Bayern Munich and Bayer Leverkusen through, Schalke join last season's finalists Borussia Dortmund in the last 16 as all four Bundesliga sides progressed to the knock-out phase.