Murphy netted his sixth Europa League goal to overtake previous record holder Stevie Kirk and give Motherwell the edge after the first leg of the third qualifying round.
However, Alexander Mathisen's last-gasp penalty denied Motherwell what would have been their greatest European result and kept the tie finely balanced ahead of next Thursday's return leg at Fir Park.
Murphy fired home from close range three minutes after the restart as Motherwell asserted themselves following an even first half.
The Steelmen had enjoyed the better of the first-half chances with Tom Hateley and Stevie Hammell both going close, although the home side threatened from set-pieces and hit the woodwork.
Brown's men looked like they had survived some late pressure to record their third successive victory by the only goal in their European campaign, but a foul by Steven Saunders allowed the hosts to level in a cruel finale.
Brown named the same team that won away against Icelandic league leaders Breidablik last Thursday as they stepped up a level to face a side sitting fourth in the Norwegian top flight after 18 games.
The home side forced several early corners, but Well managed to restrict the space around 6ft 8in home striker Tor Hogne Aaroy.
Hateley came close in the 12th minute with a stunning 30-yard volley that Anders Lindegaard tipped over but Brown's men soon survived a scare when Jamaica winger Demar Phillips struck the junction of bar and post from similar range.
The home side were throwing balls into the box but Motherwell were next to threaten in the 27th minute when Hammell's shot was tipped over following a one-two with John Sutton.
Murphy headed just wide from Hateley's near-post cross but Aalesund controlled possession for much of the latter stages of the half, with the Steelmen surviving more set-pieces.
Brown made a double substitution at half-time with Sutton, who required treatment late in the first period, and Ross Forbes making way.
Chris Humphrey and Bob McHugh came on and the substitutes were instrumental as Motherwell made a lightning start to the second half.
Humphrey's driven ball across the face of goal caused panic in the home defence before McHugh's deflected shot just cleared the bar.
And Motherwell soon struck when Humphrey cut back Hateley's back-post cross for Murphy to fire home from eight yards.
The home side tried to hit back immediately but Darren Randolph held Johan Arneng's strike from the edge of the box.
Saunders almost gifted the Norwegians an equaliser when he was short with an attempt to chest the ball back to Randolph, but Magnus Olsen shot over after intercepting.
The former Charlton goalkeeper was looking assured as Motherwell faced an aerial bombardment from wide areas, and he was grateful to see skipper Stephen Craigan's unorthodox clearance just clear his own bar.
Phillips, who had a brief spell with Stoke, blasted a free-kick just over as Motherwell continued to live dangerously.
Randolph was called into action in the 81st minute when he blocked Aaroy's shot with his feet and Motherwell looked set for a memorable victory when the keeper held a header from the giant striker in the closing minutes.
But Saunders fouled Phillips in injury-time before substitute Mathisen found the top corner from the spot.