After a delightful one-two with Jason Roberts in the 73rd minute at Fratton Park on Sunday, Rovers’ Paraguayan striker claimed his sixth goal in the last seven games.
Santa Cruz left Jamie Ashdown – standing in for David James – helpless after the goalkeeper had seconds earlier punched away from the same player.
There was little meaningful action in the first half, but Jermain Defoe had three efforts for Pompey and Ashdown had to save well from Roberts at the other end.
The second half was just as uneventful but Sulley Muntari scuffed a chance to put Pompey ahead just a minute before Santa Cruz settled things.
Pompey must now beat Cardiff in the FA Cup final next month to get into Europe.