The duo, who took 19 out of the 20 wickets to fall -- the other was a run-out -- dismissed Pakistan for 222 in their second innings after England had piled up 461 for nine declared.
Fast bowler Harmison, who had taken six for 19 in Pakistan's meagre first innings, polished off the tail on his way to a second innings return of five for 57.
His match figures of 11 for 76 were the first time he had taken 10 wickets in a Test.
And it was the first time any England bowler had taken 10 or more in an Old Trafford Test since Jim Laker, 50 years ago this week, returned world record match figures of 19 for 90, against Australia.
Harmison, who'd wrapped up the match with a devastating spell of four wickets for 13 runs in 22 balls, felt he was getting back to his best after missing the early part of the season with a shin injury.
And the 27-year-old Durham quick was also grateful to Lancashire grounsman Peter Marron for producing a hard pitch with pace and bounce.
"Obviously I saw the pitch on the first day and I was glad," said Harmison.
"I feel the more I bowl the more consistent I get and the better I get."
Harmison also praised the contribution of Panesar. "He was awesome - Monty was brilliant and every time he puts the England shirt on he gets better and better," the paceman told Sky Sports.
It was Panesar, with a Test-best five for 72, who ripped through Pakistan's top-order second time around, the left-arm spinner capturing the prize wickets of Mohammad Yousuf, Inzamam-ul-Haq and Younis Khan.
After taking three for 21 in the first innings, Panesar's match return of eight for 93 were also his best at this level.
"I was just trying to get the ball in the right area, apply pressure and stay patient," Panesar, the first Sikh to play Test cricket for England, explained. "There was turn there but I was just keeping it simple, not trying to bowl magic balls."
Pakistan coach Bob Woolmer had no complaints about the result, even though his side, as in the drawn series opener at Lord's, were without leading fast bowlers Shoaib Akhtar (ankle), Mohammad Asif (elbow) and Rana Naved-ul-Hasan (out of the whole tour with a groin injury).
"We can't make excuses, we just played pretty poorly," Woolmer said.
"But I thought Harmison bowled very well. He bowled beautifully on the first morning. He was the big difference. He got steep bounce right at the beginning.
"But 119 all out is not good enough on the first day of any Test match."
Pakistan were in deep trouble at tea on 167 for five.
And 14 balls after the resumption, Panesar had his fifth wicket of the innings when Faisal Iqbal (29) edged a delivery that turned and bounced straight to Marcus Trescothick at first slip - a classic spinner's wicket.
Harmison then took charge, having Shahid Afridi taken by England captain Andrew Strauss at second slip and, two balls later, seeing Mohammad Sami caught behind - one of five dismissals in the innings for wicket-keeper Geraint Jones who was playing with a broken finger sustained on Friday.
The tail crumbled and when Abdul Razzaq (13) skied a simple catch to Jones Pakistan's resistance had ended inside 68 overs.
Panesar struck with the first ball after lunch, removing the dangerous Yousuf for 15 after he drew the batsman forward and beat the outside edge before Jones completed the stumping.
Then 101 for three became 117 for four when Panesar saw off Inzamam for an unlucky 13.
The Pakistan skipper played the ball onto his toe as he pushed forward and Alastair Cook took the catch at silly point.
Panesar had then taken two wickets for nine runs in 18 balls.
Younis meanwhile completed a 91-ball fifty with six fours.
But 24-year-old Northamptonshire bowler Panesar was unruffled and removed Younis lbw for 62 when the batsman played no stroke.
Pakistan resumed Saturday on 12 without loss, and soon lost makeshift opener Kamran Akmal for four, caught behind by Jones off a Harmison delivery that rose alarmingly off a length.
Strauss, captaining England in the ongoing absence of Andrew Flintoff with an ankle injury, brought Panesar into the attack in the 11th over of the day to start an unchanged spell of 27 overs.
Panesar was soon getting sharp turn, as well as bounce, and his accuracy was rewarded in his fourth over when Farhat, on 34, lunged forward and gave a simple bat/pad catch to Ian Bell at short leg.
Cook (127) and Bell (106 not out) had earlier built on Harmison and Panesar's first innings work by batting England into a commanding position with their second successive Test hundreds after they had each reached three figures at Lord's.
The third Test starts at Headingley on Friday and former England batsman Woolmer admitted: "We could lose the series at Headingley so it's a cup final for us there now."