The defender put Spurs in the lead with a header in the 74th minute before Gareth Bale added a second for Tottenham three minutes later.
Bolton's Kevin Davies scored in the 90th minute to give the away side some hope, before Louis Saha closed out the match to make it 3-1.
It was Nelsen's fourth appearance for the club since he joined from Blackburn on deadline day in the January transfer window. He scored just eight goals in 172 matches for his previous club.
Tottenham will now play Chelsea in the semifinals as they look to reach the FA Cup final for the first time since 1991 when they won the last of their eight titles.
Muamba, whose heart stopped beating on its own for 78 minutes after collapsing on March 17, was visited by members of the Bolton team before the match.
In today's other match, Everton beat Sunderland 2-0 in a quarterfinal replay at the Stadium of Light, a cool Nikica Jelavic finish in the first half and David Vaughan's own goal on 57 minutes sending the visitors through to a derby semifinal against Liverpool.
