Reliever Brian Moran was also claimed off waivers by the Miami Marlins.
Davis was placed on the injured list for undisclosed reasons and has not made an appearance this season. In 37 games last year, the 28-year-old slashed .181/.266/.265/ with two home runs and six RBI.
In two games this season, Gaviglio struggled with a 27.00 ERA and a strikeout in one inning of relief. Moran was signed by the Blue Jays to a one-year deal this past off-season and recorded a strikeout in one inning of relief.