Baningime was dismissed for a studs-up challenge on Roland Idowu after just 15 minutes, with the match still goalless at Tynecastle.
But after grinding out a 1-0 win with 10 men against Dundee on Sunday, Hearts once again took maximum points despite their numerical disadvantage.
Second-half goals from captain Lawrence Shankland and substitute Tomas Magnusson left Derek McInnes' side six points clear of third-placed Rangers and second-placed Celtic, who they host in their next league match a week on Sunday.
Benjamin Nygren's 10th league goal this season extended Celtic's revival under Martin O'Neill.
The Swede headed home an Arne Engels corner two minutes before half-time.
Celtic had to survive a scrappy performance at the Falkirk Stadium before emerging with their second clean sheet in two games since O'Neill returned for a second spell as interim boss this season.
O'Neill replaced Wilfried Nancy, who was sacked after a disastrous eight-game reign, and masterminded a 4-0 win over Dundee United last weekend.