Goals from Josh Rees and Louis Dennis in either half ensured the visitors followed up Saturday's goalless draw at Eastleigh in impressive fashion.
Rees opened the scoring in the 35th minute, the midfielder slotting home from Dennis' cutback, before the latter added a second in the 70th minute on the counter-attack after Ryan Bird's header was cleared off the line at the other end.
Dover set up a tense final 15 minutes when David Gregory bundled the ball into his own net following substitute Kadell Daniel's shot from a free-kick.
Match report supplied by the Press Association.
Match ends, Dover Athletic 1, Bromley 2.
Second Half ends, Dover Athletic 1, Bromley 2.
Substitution, Bromley. Alan Dunne replaces Louis Dennis.
Ryan Bird (Dover Athletic) is shown the yellow card.
Substitution, Dover Athletic. Mitchell Pinnock replaces Mitch Brundle.
Substitution, Dover Athletic. Tobi Sho-Silva replaces Manny Parry.
Own Goal by David Gregory, Bromley.  Dover Athletic 1, Bromley 2.
Brett Williams (Bromley) is shown the yellow card.
Goal!  Dover Athletic 0, Bromley 2. Louis Dennis (Bromley).
Iffy Allen (Bromley) is shown the yellow card.
Ben Chorley (Bromley) is shown the yellow card for a bad foul.
Substitution, Bromley. George Porter replaces Adam Mekki.
Substitution, Bromley. Iffy Allen replaces Luke Wanadio.
Second Half begins Dover Athletic 0, Bromley 1.
First Half ends, Dover Athletic 0, Bromley 1.
Connor Essam (Dover Athletic) is shown the yellow card.
Goal!  Dover Athletic 0, Bromley 1. Josh Rees (Bromley).
First Half begins.
Lineups are announced and players are warming up.
Bromley claimed their first National League win of the season courtesy of a 2-1 victory at Dover.