After an uneventful first half, Rigg slotted home the visitors' only real chance of the second half to secure their place in round three, as Wycombe's recent free-scoring exploits deserted them at a crucial time.
The absence of loanees Alfie Mawson, Danny Rowe and Fred Onyedinma, plus the suspended Paris Cowan-Hall, left Gareth Ainsworth with a depleted squad which meant Sido Jombati moved to centre-half with Jose Scowen slotting in at right-back, paving the way of Max Kretzschmar and Sam Wood to come into the midfield.
Just as they’d started slowly in the first round at Barnet, the reshuffled Wanderers were almost behind within a minute when Matt Tubbs looped a header just over the crossbar, and the same player was almost sent through 1-on-1 moments later before Joe Jacobson recovered just in time.
The hosts gradually settled and almost opened the scoring from an inventive free-kick routine, which saw Jacobson’s delivery spark panic before Kretzschmar’s low drive took a deflection and needed a smart save from keeper James Shea.
Jacobson was again the architect for the next chance, as his free-kick fell to Paul Hayes but volley from a tight angle was held at the near post by Shea.
Aaron Pierre was shown a yellow card for simulation even though he was clearly obstructed just outside the box after a run from his own half, and the chances then dried up for both sides as the game developed into a cagey midfield battle.
Matt Tubbs as flagged offside just before slotting the ball into an empty net after the interval, before the deadlock was broken courtesy of wideman Sean Rigg.
Wimbledon worked the ball into the box neatly and Rigg found himself in space to calmly tuck the ball into the corner, giving a previously untested Ingram no chance.
Wanderers responded by bringing Aaron Holloway and Hogan Ephraim into the fray and weren’t far from an equaliser when Sam Wood’s rocket from 30 yards stung the palms of James Shea and bounced to safety.
Moments later, Peter Murphy almost got on the end of a bouncing ball down the middle but again the visitors survived as Wycombe looked to crank up the pressure.
Josh Scowen – who scored in the league meeting in October – drilled wide with his weaker foot on 81 minutes after being allowed to reach the edge of the D without a challenge, but the visitors’ resolve was standing firm as the game reached its closing stages.
Six minutes of added time gave Wycombe hope but Aaron Holloway sent a header over the bar as Ainsworth's men were unable to keep their cup hopes alive and they'll look to continue their impressive form in the league when Southend come visiting next week.
Wycombe: Ingram, Scowen, Jacobson, Jombati, Pierre, Kretzschmar (Ephraim 70), Bloomfield, Murphy, Wood (McClure 83), Craig (Holloway 66), Hayes.
Unused subs: Richardson, Lewis.
Wimbledon: Shea, Fuller, Kennedy (Harrison 89), Goodman, Barrett, Francomb (Pell 77), Bulman, Moore, Rigg (Azeez 77), Akinfenwa, Tubbs.
Unused subs: McDonnell, Bennett, Sutherland, Oakley.
Referee: Mr C Kavanagh
Attendance: 3,196 (1,035 from Wimbledon)
Winning 50/50 Number: 154171
Prize Money: £551.00