Ritchie restored Bournemouth's lead with a key second goal 10 minutes into the second half of a match they dominated without turning their superiority into goals until late on.
When Brett Pitman fired Bournemouth into an eighth-minute lead it looked as though the south-coast play-off contenders might cruise to a comfortable win.
But cash-stricken Bury, who named only four substitutes, were competitive opponents and when Craig Jones pounced on a slip by home defender Dan Seaborne midway through the first half, the dangerous Andy Bishop headed the visitors level.
Bournemouth were ultimately too strong though and once Ritchie had restored their lead with a cracking shot from the edge of the box, the outcome was seldom in doubt.
Bournemouth scored twice more in the last 10 minutes - Harry Arter rifling a shot into the roof of the net and then Matt Tubbs scoring the fourth after stepping off the substitutes' bench.