The home side were more purposeful in possession than their Norwegian visitors during a comfortable first half and stood firm in defence when put under concerted pressure after the interval.
Substitute Jonathan Page netted a late third to round off an impressive victory which sends Motherwell into a potentially lucrative tie - one step from the group stages and a round further than they achieved last season.
Murphy had broken Motherwell's European goalscoring record in the first leg but his side were denied a memorable win when Alexander Mathisen levelled from the spot with the last kick of the ball.
However, it took less than four minutes for Murphy to cancel out the penalty as Craig Brown's decision to replace Ross Forbes with pacy winger Chris Humphrey paid off handsomely.
The Englishman burst past his man to latch on to Steven Saunders' throw down the right wing before cutting the ball back into the visitors' penalty box. Murphy was unmarked and sent a first-time effort into the top corner from 16 yards.
Motherwell extended their advantage after some patient build-up play in front of the Aalesund back four. The ball was laid back to Saunders and the right-back's driven ball into the box was diverted home by Sutton from 14 yards.
Aalesund were lacking the creativity of injured Jamaica winger Demar Philips, who had won the penalty from Saunders in Norway and also hit the bar.
They were over-reliant on the long ball up to tall striker Tor Hogne Aaroy before the break and Darren Randolph commanded his box superbly.
The former Charlton goalkeeper comfortably held Magnus Olsen's header following a Peter Larsen cross when the visitors did carve out a chance.
Aalesund were playing three up front and pushing midfielders up to support but they could not find a way through and wide players Murphy and Humphrey were breaking forward whenever the hosts got possession.
Murphy had a decent penalty claim when he was bundled over by Trond Fredriksen's clumsy aerial challenge but Motherwell had to settle for a two-goal interval lead.
Olsen turned and fired two yards wide from long range moments after the restart and the visitors began to exert some sustained pressure.
Mathisen looked destined to head home Pablo Herrera's right-wing cross but he failed to make proper contact and Saunders nodded clear.
Motherwell were struggling to keep the ball and faced more nervous moments as their opponents, who are 19 matches into their league season, found an extra gear.
Brown switched from a back four and moved Saunders into a three-man central defence with Tom Hateley now playing at right wing-back, and the tactic helped briefly relieve the pressure. But Well were still sitting deep and Mark Reynolds needed all of his pace to prevent Aaroy latching on to a flick-on before Ville Jalasto glanced a header just wide from eight yards.
A goal for the visitors would have given Motherwell serious cause for concern and they continued to live dangerously until they killed off any lingering hopes with a third on the break in the 89th minute.
Saunders played the ball forward from the right-back position and Murphy flicked it past the only covering defender before drawing the goalkeeper and unselfishly squaring for Page to stroke into the empty net.
Page, who had replaced John Sutton five minutes earlier, was making his first appearance at Fir Park, which erupted in celebration moments later.