Jose Mourinho’s side climbed above previous leaders Liverpool on goal difference after a sterile London derby at Stamford Bridge.
Tottenham are unbeaten in nine league games and Mourinho seemed happy to frustrate his former club as they adopted a cautious approach with little attacking ambition.
Chelsea have not lost in their last 11 games in all competitions and sit in third place, two points behind Tottenham and Liverpool, who drew 1-1 at Brighton on Saturday.
Fourth-placed Leicester would go second if they beat Fulham on Monday.
Timo Werner thought he had put Chelsea ahead early in the first half with a superb finish, only for an offside flag to thwart the Germany striker.
Chelsea’s Mason Mount forced a good save from Hugo Lloris in the second half, while substitute Olivier Giroud should have done better than shoot straight at the Tottenham keeper in stoppage time.
The drab clash in west London was a stark contrast to United’s dramatic fightback from two goals down at St Mary’s.
Ole Gunnar Solskjaer’s side were facing a dismal defeat after first-half goals from Jan Bednarek and James Ward-Prowse put Southampton in control.
But Uruguay striker Cavani sparked United’s escape act after coming on at the interval.
Cavani set up Bruno Fernandes to reduce the deficit and the former Paris Saint-Germain star grabbed the equaliser from his team-mate’s deflected shot.
The 33-year-old netted again in the final moments as his third goal since signing as a free agent in October extended United’s winning to four games in all competitions.