Elijah Sindelar's 33-yard pass to Anthony Mahoungou to the Golden Gophers' 12-yard line set up the Jones sweep of right end. Purdue iced the victory with 10 seconds remaining when linebacker Ja'Whaun Bentley returned an interception 76 yards for an insurance touchdown.
Minnesota had taken a 17-16 lead with 2:26 to go on Emmit Carpenter's 38-yard field goal, capping an 18-play, 62-yard drive.
J.D. Dellinger had kicked a 19-yard field goal with 9:58 left to give the Boilermakers a 16-14 lead.
The game was delayed by heavy rain for one hour and 28 minutes, just a few moments after the Dellinger field goal. A crowd of 42,085 scattered to the corridors as the heavy rain began to fall.
It was the first Ross-Ade Stadium weather delay since a 52-6 victory against Eastern Illinois on Sept. 8, 2007.
Purdue sophomore quarterback Sindelar, who is battling junior David Blough for the starting job, completed 19 of 26 passes for 248 yards and one touchdown.
The Boilermakers (3-2, 1-1 Big Ten) sliced an eight-point halftime deficit to 14-13 with 10:38 remaining in the third quarter on Sindelar's 22-yard touchdown pass to D.J. Knox, completing a 10-play, 52-yard drive.
Purdue grabbed an early 6-0 lead, taking advantage of an 11-yard punt, setting up a 20-yard touchdown pass from Blough to tight end Cole Herdman with 12:43 remaining in the first quarter.
The Boilermakers drove 37 yards in three plays to score their only first-half points.
Minnesota (3-2, 0-2) seized a 7-6 first-quarter lead on Conor Rhoda's 3-yard touchdown pass to Tyler Johnson at the 4:56 mark, five plays after safety Adekunle Ayinde intercepted Blough at the Purdue 39-yard line.
The Golden Gophers increased their first-half lead to 14-6 on Rhoda's 3-yard touchdown pass to tight end Brandon Lingen with 12:13 remaining in the second quarter.
Kamal Martin's interception of Blough in the end zone set up a five-play, 80-yard drive that included Rodney Smith's 51-yard run to the Purdue 10.
The Boilermakers turned the ball over four times during the first half, including lost fumbles by wide receiver Terry Wright and Sindelar.