Ebrima Singhateh put Sparta ahead in the 35th minute with a fine solo effort, but the hosts' position deteriorated when Tobias Guddal was sent off two minutes later for a second yellow card.
Bournemouth had already missed a penalty through Lewis Cook, but took advantage of their numerical superiority after the break. Adams equalised in the 57th minute before Evanilson put the visitors ahead nine minutes from time. Rayan completed the scoring in the 90th minute.
Sparta coach Brian Priske was proud of his players' first-half display, despite the eventual defeat.
"The players did really well in the first half and we were the best team until the red card," he told UEFA.
"It was obvious it was going to be difficult from there on. We did well until we conceded the goal, but I have to praise my players who dominated a strong Premier League side for a long period of time."
Priske also regretted the manner in which Bournemouth scored, pointing to mistakes in possession.
"We knew we would suffer in the second half, but we tried hard to be compact. It is a pity all three goals came from ball losses from our side, one or two of them were too easy. At the end, Bournemouth showed they are very good Premier League team."
He was particularly pleased with Singhateh's opener, which came from a long pass by the goalkeeper.
"It was a brilliant goal, we know what Ebrima Singhateh can offer us in these transitions. It was perfect pass from the goalkeeper and [good execution] from Singha to finish it."
Bournemouth manager Marco Rose was pleased with his team's response after a difficult opening period.
"We missed a penalty and conceded a goal, but our reaction in the second half was really good," he said.
"We knew we had a good starting 11 and a really good bench to make changes. It is a really good sign for the squad."
Rose added: "No clean sheet but we won and this is the most important thing. [It was] a difficult game against a good opponent. They made it really hard for us to get it into our game. We struggled a little bit in our combinations at the beginning but we were ready and we had a good attitude."
The Bournemouth boss hopes his side can carry their European progress further.
"We would like to go as far as possible. We enjoy the moment, we enjoy being in Europe and our fans travelled with us again. The first thing is we want to go to the [knockout phase] and then we will see."