The striker marked his first game as Irish skipper and the first game of Steve Staunton's reign as manager by netting the second goal at Lansdowne Road.
Damien Duff opened the scoring in the first half, Robbie ran on to Ian Harte's header and lashed home for 2-0 a couple of minutes into the second half and Liam Miller completed an impressive night's work in Dublin.
It was record goalscorer Robbie's 26th goal in 65 appearances for the Republic.
The only disappointment from a Spurs point of view was that Stephen Kelly remained on the bench as an unused substitute and still waits for his first senior cap.
"I thought it was a tremendous performance," said Robbie afterwards. "The lads showed real team spirit out there and it's always nice to start off with a good result."
New captain Robbie Keane led by example with a goal as the Republic of Ireland beat Sweden 3-0 in Wednesday night's friendly.