Geis fired in a 51st minute penalty, with the 22-year-old making his return after being banned following a violent tackle on Borussia Moenchengladbach's Andre Hahn on Oct. 25.
Dutch striker Klaas-Jan Huntelaar doubled their lead in the 73rd and Franco Di Santo scored their third, a minute after Hanover had pulled a goal back with French teenager Allan Saint-Maximin.
Schalke now have 24 points, one behind third-placed VfL Wolfsburg who play on Saturday against second-placed Borussia Dortmund, with Hanover in 14th.
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The hosts, who had failed to win any of their league games during Geis' absence, were in command but struggled to get past Hanover's defence, relying mainly on rare solo efforts from Huntelaar and teenager Leroy Sane.
Sane, whose sensational season earned him his first senior Germany cap last month, had their biggest chance on the stroke of halftime with keeper Ron-Robert Zieler, in his club record 157th consecutive league game, keeping his cool in their one-on-one to deny the 19-year-old.
Schalke's pressure eventually paid off early in the second half when Huntelaar was brought down in the box and Geis confidently converted the spot kick.
Huntelaar was rewarded with his fifth goal of the season when found a gap in the defence and volleyed in from eight metres.
Hanover showed some sign of life with substitute Saint-Maximin's 20-metre missile but Argentine Di Santo took the wind out of their sails with a quick counterpunch and his first goal for Schalke.
Undefeated Bundesliga leaders Bayern Munich, top on 40 points having won 13 of their 14 league games, travel on Saturday to in-form Gladbach, unbeaten in nine games under coach Andre Schubert.
(Reporting by Karolos Grohmann, editing by Alan Baldwin)
