The game was finely poised at 23-23 at the break but heavy rain affected both sides' play in the second half.
Flanker Tom Rees ran in two tries for Wasps with Tom Varndell and Harry Ellis answering for Leicester, but the match was ultimately decided by the kickers.
Mark van Gisbergen kicked Wasps into the lead with 14 minutes left but Goode's fifth penalty snatched a draw.
Leicester coach Pat Howard: "We wanted to win but I'm happy we fought back when we went behind.
"We took our opportunities and kept our composure. We're OK with the result. There's plenty to work on but it was a difficult game to handle.
"There were plenty of mistakes and it was not the sort of weather either team expected when we started the day. Blokes were slipping all over the place."
Wasps: Van Gisbergen; Sackey, Waters, Abbott, Voyce; King, Reddan; Payne, Ibanez, J Dawson; Shaw, Birkett; Rees, O'Connor, Worsley (capt).
Replacements: Hoadley for Abbott (78), M Dawson for Reddan (52), Gotting for Ibanez (44), Lock for Shaw (72), Hart for Rees (48), Bracken for Gotting (55), Brooks for Bracken (79).
Leicester: Murphy; A Tuilagi, Lloyd, Gibson, Varndell; Goode, Ellis; Holford, Chuter, Moreno; Cullen, Kay; B Deacon, Jennings, Corry (capt).
Replacements: Healey for Murphy (40), Broadfoot for Gibson (77), Taukafa for Varndell (71), Cornwell for Ellis (40), White for Moreno (40), L Deacon for Cullen (50), Abraham for B Deacon (66).
Sin Bin: Jennings (28).
Att: 8,459
Referee: David Pearson (RFU).