"Tom does a great job of getting the ball to the open guy," Patriots coach Bill Belechick told reporters.
New England led 21-10 late in the second quarter on Wes Welker's second touchdown reception before Dallas rallied.
Cowboys quarterback Tony Romo hit Terrell Owens with a 12-yard touchdown pass with 46 seconds left in the half to close the gap to 21-17.
Then Romo, who completed 18 of 29 passes, found Patrick Crayton for an eight-yard score and a 24-21 Dallas lead with 10:20 to go in the third quarter.
New England was soon back on top at 28-24 on a 1-yard Tom Brady to Kyle Brady pass play. Stephen Gostkowski's 45-yard field goal made it 31-24 before Stallworth sealed the win.
"They are an outstanding team with tremendous match-up problems for any defense," Cowboys coach Wade Phillips told reporters.
"We wanted to be one of the elite teams but obviously we are not."
Stallworth caught seven passes for 136 yards from Brady with Welker grabbing 11 for 124 yards.
Randy Moss finished with six catches for 59 yards and one touchdown. Owens made six receptions for 66 yards for Dallas.
*The New Orleans Saints won their first game of the season, a 28-17 victory over the Seattle Seahawks, with Drew Brees passing for 246 yards and two touchdowns.
*Rookie Adrian Peterson rushed for 224 yards and three touchdowns in the Minnesota Vikings' 34-31 road win over the Chicago Bears. Ryan Longwell kicked a 55-yard field goal as time expired to clinch the win.
*LaDainian Tomlinson rushed for four touchdowns and 198 yards in the San Diego Chargers' 28-14 home win over the Oakland Raiders.
*David Garrard threw for two touchdowns and Maurice Jones-Drew rushed for two more as the Jacksonville Jaguars rolled to a 37-17 victory over the Houston Texans.
*The Green Bay Packers (5-1) rallied to beat the Washington Redskins 17-14 with Charles Woodson returning a fumble 57 yards in the third quarter for the go-ahead touchdown. Packers quarterback Brett Favre broke the NFL record for interceptions with his 278th as the Redskins picked off two of his passes.