Bournemouth were looking to progress past the third round for the first time in four seasons.
Manager Eddie Howe made nine changes from the 4-0 defeat at West Ham on New Year's Day which had left the injury-hit Cherries in the relegation zone.
Luton's form was equally poor, with the Hatters having won just three of their 15 previous matches.
Harry Wilson had already come close to opening the scoring when Billing put Bournemouth in front with only eight minutes on the clock.
Diego Rico floated in a cross from the left and Billing sent a volley spinning into the net.
Solanke, who joined Bournemouth from Liverpool a year ago to the day, thought he had doubled the hosts' lead by ending his goal drought.
But the striker's header was ruled out after VAR adjudged Harry Wilson was offside when Ryan Fraser played in a corner.
The home side continued to dictate possession in Luton's half but, as has been the story of their plummet down the Premier League table, lacked a cutting edge.
The wasteful Cherries should have been made to pay in first-half stoppage time when Billing fouled Matty Pearson in the penalty area.
But Sheehan was unable to take advantage as he sent the resulting spot-kick crashing against the crossbar.
With 63 minutes on the clock, the pace had slowed once more only for Billing to once again briefly wake the home fans with a wayward shot from distance, prompting Howe to send for substitute Callum Wilson.
The change instantly paid dividends, with the England striker scoring with his first touch.
Travers played a long ball out from the back that let Harry Wilson run at the Hatters' defence and, despite a heavy touch, he was able to centre for Callum Wilson to confidently finish into the bottom corner with his right foot.
Billing made sure of the win with 11 minutes remaining with a deflected 25-yard shot before Solanke crowned the victory three minutes later by smuggling the ball home from inside the box.