Robshaw, who led England in every game of the RBS 6 Nations, was everywhere for Quins, and it was his second-half try that proved decisive in the end.
Quins had taken the lead after just three minutes through a Nick Evans penalty and followed that up with a try from Tom Williams after a break from George Robson.
Leicester finally got themselves on the scoreboard with a penalty from teenage fly-half George Ford but Evans cancelled that out to make it 11-3 after 22 minutes.
Ford added another penalty before an overthrown Quins lineout proved very costly with Dan Cole finding Steve Mafi who sprinted from halfway to dive over, Ford's conversion giving Leicester the lead for the first time.
However with a minute to go before the break Thomas Waldrom was sent to the sin bin as he came in from the wrong side and Evans slotted the penalty to send Quins in 14-13 up at the break.
Conor O'Shea's men took full advantage of the extra man after the break with two penalties from Evans before Robshaw ran the perfect line to crash over from close range.
Evans added a sixth penalty of the day to stretch the lead to 17 and the game looked up for Leicester.
But having made the last eight finals, the Tigers weren't about to give up the ghost and a scintillating break from Ben Youngs set up Anthony Allen for a try under the posts.
Ford converted, and added another penalty soon after to get Leicester back within a converted try with ten minutes remaining.
But despite a late lineout deep in Quins territory, Robshaw and co held out to record a historic victory for the Stoop outfit.