Martin Boyle's early goal was enough to give Paul Hartley a winning start as Dundee manager as the Dark Blues went top of the Championship with a 1-0 home win over nine-man Hamilton Accies.
The visitors, who stay third, had both Stephen Hendrie and Ziggy Gordon sent off early in the second half at Dens Park.
Falkirk, who started the day as leaders, went down 2-0 to fourth-placed Queen of the South at Palmerston, where substitute Gavin Reilly scored a late double for Jim McIntyre's men.