Lambert's low free kick six minutes from time took two deflections but Rovers, who knocked out Premier League Fulham in the third round and earlier had a Lambert header disallowed for a push, were well worth their home victory.
Lambert, celebrating his 26th birthday on Saturday, provided the highlight in a game that was short of chances as Rovers reached the quarter-finals for the first time in 50years.
The stand-out game of the round takes place later on Saturday (5:15p.m.) when Manchester United and Arsenal, the two most successful clubs in FA Cup history, meet at Old Trafford.
Thereare four other matches on Saturday, including holders Chelsea against Huddersfield, with a further two on Sunday.
(Reporting by Mitch Phillips, Editing by Trevor Huggins)