The Latics have bagged former Newcastle defender Tavernier, who impressed on loan at Rotherham last season, and Osasuna striker Riera as they look to build a side that can take them back to the Premier League.
Rosler told the club's official website: "James is a young player with experience and in that respect he fits into the type we want to bring here in that he is young, vibrant, full of energy and determined to progress with us.
"In Oriol, we have a proven goalscorer who has signed for us within our pay structure in the Championship and for the right fee.
"It was a complicated deal which took a lot of negotiation and the chairman also played a pivotal role."
The Latics manager admitted the club may not have sealed the deals had it not been for current player James Perch and recently-departed midfielder Jordi Gomez.
Perch and Gomez knew Tavernier and Riera from their time at Newcastle and Barcelona B respectively.
Rosler added: "Tavernier knows Perchy well and I know he spoke very well about his experience here and I also want to thank Jordi for the role he played in Oriol moving here.
"Oriol and Jordi were team-mates at Barcelona and I think it shows you the regard that Jordi has for our club that he would help us in that way by talking to him and telling him about the benefits of signing for us.
"Overall, I am delighted with both signings. They definitely add something to the squad that we did not have previously and I am looking forward to working with the pair of them."