The visitors put in a battling performance despite having Jeff Hendrick sent off for a two-footed challenge on Jose Holebas after six minutes and falling behind soon after when Troy Deeney headed home Niang's cross.
The Frenchman turned scorer on the strike of halftime, heading in from Holebas as Watford took control although they had to negotiate a nervous last 10 minutes after Sebastian Prodl handled on the line to give Burnley a penalty, which Ashley Barnes converted.
The win, on the back of last week's 2-1 success at Arsenal, was the first time Watford have posted two successive victories since September. For Burnley, record signing Robbie Brady made his debut as a second-half substitute, along with another January arrival, Ashley Westwood.
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(Reporting by Neil Robinson, editing by Ed Osmond)
