Porter had 15 of his 19 points in the second half, including 10 straight over the final 5:42, and grabbed 14 rebounds for the Hoyas (17-4, 7-3 Big East).
Eli Carter had 23 points for the Scarlet Knights (12-10, 3-8), who lost their sixth straight thanks to Porter's late domination.
"Otto Porter is Otto Porter, that's what we've come to expect from him," Georgetown coach John Thompson III said. "I thought he was terrific. I always think he's terrific. We ran a couple plays to get the ball in his hands and he came through.
"He got two big rebounds when we needed him to. He's one of the best players in the country and the best players in the country come through when it's winning time. He's consistently come through when it's winning time."
With just under 6 minutes remaining Rutgers led 58-57 before Porter hit a 3-pointer to start his personal 10-point run. Rutgers tied it on two free throws by Mike Poole, which would complete a string of nine straight scoring sequences that produced either a lead change or a tie.
Porter then scored on a fast-break layup to make it 62-60 with 4:44 left and give the Hoyas the lead for good. With Georgetown leading 62-61, the Hoyas came out of a timeout and got a three-point play from Porter off an entry pass from Starks to lead by four with 1:15 remaining.
"We're in the 1-and-1, we wanted to get inside there," said Porter, a sophomore forward. "I got it inside and they fouled me.
"It's the last 5 minutes of the game. We want to stay with our stuff and get the ball inside."
Carter scored a quick basket to bring Rutgers within 65-63 with 55 seconds left, but Porter responded again by hitting both ends of a 1-and-1. The Knights missed two straight shots and Hopkins made two foul shots with 14 seconds left to clinch it.
Hopkins was 6 for 9 from the field and scored 12 points in the second half.
"With Nate (Lubick) getting into foul trouble, coach allowed me to play a little bit longer, and with that being said I was able to get in my groove," Hopkins said. "When I made mistakes he let me play through it."
After both teams shot well in the first half, Georgetown maintained its pace in the second while Rutgers cooled off.
Georgetown shot 57.4 percent overall (27 for 47) and hit 60.9 percent (27 for 47) in the second half. Rutgers shot just 38.9 percent (21 for 54) for the game and the Scarlet Knights were 10 for 33 (30.3 percent) after intermission.
"In the Big East you just have to find a way to win," Carter said. "In the end, they found a way to win with Porter cutting and running their offense and just making a tough, tough shot and hitting the 1-and-1 to top it off."
Neither team could take control as the game featured 14 lead changes and 11 ties. The biggest lead by either side was seven points by Georgetown in the first half.
"This was a typical Big East game," Thompson said. "I told the guys at halftime this is going to be one of these games where you just have to grind it out, possession by possession."
The Knights' biggest problem, as it has been during their losing streak, was not converting offensive rebounds. Coach Mike Rice said the Knights had 21 misses in the paint.
"Georgetown is a very talented team, they don't beat themselves," Rice said. "You have to go out and create those opportunities and take advantage. At times we did but in Big East play you have to do it down the stretch and we didn't."