Both sides went into the match on the back of victories in the opening round, with the Chiefs having won 42-15 away to newcomers London Welsh and the Ospreys beating Gloucester 33-27 at home.
The Chiefs gave a debut to hooker Jack Yeandle, who was joined in the front row by Luke Cowan-Dickie and Hoani Tui.
In the backs Nic Sestaret switched to the centre, with Jack Nowell named on the wing and Myles Dorrian moving to full-back.
The Ospreys also made changes with Owen Williams at tighthead prop, Ian Gough and Sam Williams in the second row and Arthur Ellis at openside flanker, while Sam Davies was at fly-half and Tom Habberfield at scrum-half.
England Under 20 fly-half Henry Slade kicked the Chiefs into an early lead with a 22-metre penalty, but an error from Exeter captain Junior Poluleuligaga led to Ospreys scoring the opening try through number eight Morgan Allen from close range which was safely converted by Davies.
Slade missed a penalty before the Chiefs scored their first try with 12 minutes of the first half remaining.
The home side were initially stopped short of the line, but when the ball was recycled lock Will Carrick-Smith found his way over the line to claim the touchdown.
Slade added the simple conversion but Ospreys were quickly back on level terms with a long range penalty from fly-half Davies.
Slade missed another penalty just before the break, but the Chiefs kept pressing for the second try and Myles Dorrian picked up from the ruck and slid over the line with Slade adding the extras.
Davies was wide with a 30-metre drop goal attempt, leaving the Chiefs 17-10 ahead at half-time, but he soon made amends in the second half with a penalty.
Left-wing Mark Foster then dropped the ball with the line at his mercy but, thankfully for the Chiefs, the referee pulled play back for a penalty for offside in front of the posts and Slade restored a seven-point advantage for the home side.
Slade stretched the lead to 10 points with another penalty before the Chiefs had to play the final five minutes with 14 men when, having used all their replacements, Dave Ewers had to be helped from the pitch with a leg injury.
Chiefs replacement hooker Neil Clark was sent to the sin-bin in the closing moments but replacement fly-half Matthew Morgan missed a penalty that would have earned the Ospreys a losing bonus point.
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.