Robshaw, 34, led his team out for his 300th and last appearance and played the full match as tries from Alex Dombrandt, Joe Marchant (two) and James Lang, along with 12 points by fly-half Brett Herron, saw the Londoners just get home.
But, after one of their worst seasons in living memory, second-bottom Leicester had pride to play for and they nearly spoilt Robshaw's party.
The Tigers had Quins on the back foot in the early exchanges and soon there were plays that number eight Dombrandt and Leicester's England scrum-half Ben Youngs will remember for different reasons.
Leicester flanker Tommy Reffell almost scored from a five metre line-out but fell a metre short and had Dombrandt's big hand underneath the ball anyway.
The home side would not be denied, though, and Tom Youngs found his way to the line from a maul, although the conversion, which should have been simple for Ford, was inexplicably missed.
Then, Ben Youngs tried to intercept a loose pass outside the Quins 22 but dropped the chance.
Moments later, the number nine became the victim as he launched a pass inside opposition territory only to see Dombrandt snatch it in mid-air and run 70 metres to score in the corner.
Herron converted to give Quins the lead for seven minutes before Ford landed two long-range penalties.
It was end-to-end stuff as Herron landed a routine penalty and wing Marchant superbly got past Nadolo to squeeze home for a corner try which Herron converted as the visitors earned a 17-11 half-time lead.
Nadolo responded with a try of his own five minutes into the second half, just keeping his foot in touch as Marchant tackled him, and Ford's superb touchline conversion gave Leicester a one-point lead for the third time in the game.
That advantage did not last long, though, as a great team effort saw Marchant go over on the right for his second try, converted by Herron.
And another well-worked Quins try in the 53rd minute, with Lang finishing it off and securing the try bonus point, turned the game as the Londoners went 11 points ahead.
But the result was in the balance to the end. Ford and Herron swapped penalties before Quins prop Jordan Els was sent to the sin-bin for a silly mistake at a maul close to his own line.
It left Tigers with the man advantage, which they made the most of as centre Scott found the line for an unconverted try with 12 minutes to go.
Leicester tried to win it and debutant wing Kobus Van Wyk nearly scored in the dying minutes, but he had a foot in touch when grounding as Quins took the honours and clapped their departing star Robshaw off the field at the end.