The Londoners bore little resemblance to a side battling against relegation and could have returned home with their first Premiership victory at the Tigers' stronghold had Jeremy Staunton's kicking proved more trustworthy. Instead, they left with nothing.
The Quins fly-half – who replaced broken foot victim Andy Dunne at number 10 – missed half of his six shots at goal, while his opposite number enjoyed a flawless afternoon, finishing with 22 points.
A shock looked on the cards when early tries from prop Jon Dawson and winger Ugo Monye gave Quins a 17-9 lead after half an hour, but Leicester replied with an Austin Healey touchdown.
Goode's pinpoint kicking saw the Tigers enter half-time 19-17 up and from that point they held the edge, although Quins battled to the final whistle in a performance that will stand their Premiership survival bid in good stead.
An injury-time try from substitute Henry Tuilagi gave the score a lop-sided appearance which Leicester scarcely deserved, and Quins were unlucky not to finish with a bonus point for their spirited display.
The Tigers' England contingent came through unscathed, but back row star Lewis Moody failed to start the match after pulling out with a hamstring injury, giving the world champions another injury scare ahead of Saturday's RBS 6 Nations opener against Wales.
Quins made a dream start thanks to a costly line-out blunder by Leicester which saw hooker George Chuter fail to find his jumper, allowing blindside Nick Easter to grab the ball and drive forward.
He was halted inches short but prop Jon Dawson took up possession and burrowed his way over with only one minute and eight seconds on the clock.
Staunton added the conversion and then slotted a penalty which was sandwiched by two shots at goal from Goode, but there was more than just good kicking on show as both sides showed some neat touches in attack.
Goode landed another penalty, but Quins' second try put the Twickenham club back in the driving seat with the line-out again providing the platform for the score.
Easter caught the ball at the back of the set-piece and fed straight to Monye who, running at full tilt, broke through Leicester's midfield and skipped around two more tacklers before crossing under the posts.
Staunton improved the try which appeared to sting the Tigers into action, with Leon Lloyd carving open Quins' defence with a dummy before being caught by the visitors' cover.
Leicester continue to pile on the pressure at the ensuing line-out and a scrum, but eventually opted to take the three points through Goode when their efforts to pierce their rivals' line were frustrated.
They enjoyed more success in the 33rd minute, however, when a hanging crossfield kick from Goode was collected by England lock Ben Kay and then fed to Healey who scampered in unopposed. Goode added the extras.
Geordan Murphy raced into space as the half neared its conclusion, but sloppy handling from himself and Ollie Smith saw a good chance go begging.
Quins' hopes of claiming their first Premiership victory at Welford Road receded when Easter was sent to the sin-bin after interfering with play from an offside position and Goode collected three more points.
Evans' men repeatedly found themselves on the wrong side of the referee's whistle and this was proving costly with Goode in such fine form, sending another three points over in the 62nd minute.
Quins spent the remainder of the quarter on defensive duty as Leicester poured forward in search of a second try, and it finally came eight minutes into injury time with Tuilagi crossing in the corner.
For Leicester Tigers:Tries: Healey, TuilagiPens: Goode 6Cons: Goode 2
For NEC Harlequins:Tries: Dawson, MonyeCons: Staunton 2Pen: Staunton
Leicester Tigers: 15 Geordan Murphy, 14 John Holtby, 13 Leon Lloyd, 12 Ollie Smith, 11 Austin Healey, 10 Andy Goode, 9 Harry Ellis, 8 Louis Deacon, 7 Neil Back, 6 Lewis Moody, 5 Ben Kay, 4 Martin Johnson (captain), 3 Darren Morris, 2 George Chuter, 1 Graham Rowntree.Replacements: 16 Ephraim Taukafa, 17 Julian White, 18 Will Johnson, 19 Henry Tuilagi, 20 Scott Bemand, 21 Daryl Gibson, 22 Alesana Tuilagi.
NEC Harlequins: 15 Gavin Duffy, 14 Simon Keogh, 13 Dafydd James, 12 Mel Deane, 11 Ugo Monye, 10 Jeremy Staunton, 9 Steve So’oialo, 8 Tony Diprose (captain), 7 Luke Sherriff, 6 Nick Easter, 5 Simon Maling, 4 Roy Winters, 3 Jon Dawson , 2 Ace Tiatia, 1 Ceri JonesReplacements: 16 James Hayter, 17 Mike Worsley, 18 Simon Miall, 19 Kai Horstmann, 20 Andy Reay, 21 Henry Barratt, 22 Tom Williams
Referee: Roy MaybankTouch judges: Andy Watson, Peter HuckleAssessor: Brian Leigh