Graham, a defensive end, was placed on injured reserve and will miss the rest of the season after tearing a knee ligament.
The 32-year-old Burgess, who was cut by the Patriots before the season, was utilized by New England primarily as a defensive end in 2009, when he played 57 percent of the team's snaps and totaled five sacks.
Burgess began his career with the Eagles when he was drafted in the third round of the 2001 draft by Philadelphia.
Graham, the 13th overall pick in this year's draft, tore his anterior cruciate ligament in Sunday night's 30-27 win at Dallas.
Philadelphia sent its first-round pick (24th overall) and a pair of third-round selections (Nos. 70 and 87) to Denver to move up and select Graham. He had three sacks.
The Eagles also lost middle linebacker Stewart Bradley indefinitely. Bradley dislocated his right elbow, but coach Andy Reid says he doesn't need surgery at this time.
Information from ESPN NFL Insider Adam Schefter, ESPNBoston.com's Mike Reiss and The Associated Press was used in this report.