Frank, who took over from Australian coach Ange Postecoglou during the close season, was without injured pair James Maddison and Dejan Kulusevski, but Richarlison's double helped Spurs secure a first EPL win since April 6.
But Spurs could soon be reinforced by Crystal Palace forward Eberechi Eze, who has been strongly tipped to complete a move across London,
Mohammed Kudus twice set up Richarlison for his goals, firstly in the 10th minute before he supplied a cross which the latter volleyed in on the hour mark.
Brennan Johnson added another six minutes later to wrap up the win for Tottenham on a tough top-flight return for Burnley boss Scott Parker at his old club.
Newcastle missed the presence of Alexander Isak as they shared the spoils in a goalless draw with 10-man Aston Villa.
Isak was left out by boss Eddie Howe as the Sweden international tries to force a move away from St James' Park this summer.
But the Magpies could have done with their star man as they failed to find a way past Villa, who had Ezri Konsa sent off in the 66th minute.
Sunderland marked their return to the EPL in style after they swept West Ham aside with a 3-0 win at the Stadium of Light.
Following an eight-year absence from the league, the Black Cats went ahead just after the hour mark through Eliezer Mayenda before Dan Ballard's header doubled their advantage.
Substitute Wilson Isidor came off the bench to wrap up three points in Sunderland's first EPL appearance since May 2017.
Rodrigo Muniz struck deep into stoppage time as Fulham snatched a 1-1 draw from under the nose of Brighton at the Amex Stadium.
Brazilian forward Muniz pounced six minutes into added time after Brighton failed to deal with Harry Wilson's corner.
Albion had looked on course to make a winning start to the season when Matt O'Riley tucked home a 55th-minute penalty after Sander Berge brought down Georginio Rutter.
Bournemouth's  Antoine Semenyo, who was targeted with racist abuse by a spectator during his side's 4-2 defeat at Liverpool on Friday, has praised the backing he has been given.
Writing on social media, he said: "Last night at Anfield will stay with me forever - not because of one person's words, but because of how the entire football family stood together."
Merseyside Police said a 47-year-old man was arrested on suspicion of a racially aggravated public order offence and has been taken into custody to be interviewed.
Wolverhampton were hosting Manchester City in the late game....
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