That means the duo remained tied atop the leaderboard at 3-under-par 207. They were tied for the lead after round two and the duo shared the first-round lead with Andrew McLardy.
Peter O'Malley used a 2-under 68 to move into third place at 2-under-par 208. Peter Hanson (68) and Marc Warren (71) are in fourth at minus-1 as only five players are in red figures through three rounds.
Sergio Garcia carded his third straight even-par 70. He stands alongside Steven Jeppesen, who has shot back-to-back 69s, in sixth place.
Garcia is in the best position to keep the trophy in Spain. He is one of three Spaniards, along with Miguel Angel Jimenez and Jose Maria Olazabal, to win this crown over the first three years of the tournament.
'I hit plenty of good shots out there, but I also hit a couple of bad ones which cost me,' Garcia stated. 'I just felt like I couldn't get anything really going out there. Conditions were a little easier today overall but, in saying that, there might not have been so much wind, but I don't think I have seen wind change direction so much on a golf course in my life.'
Murphy grabbed an early lead as he drained a 7-foot birdie putt on the first. His lead was two strokes as Fasth tripped to a bogey on the first at Pula Golf Club.
Both Fasth and Murphy birdied the par-5 sixth. The next big swing came on the ninth. Murphy found a fairway bunker of the tee. He was unable to reach the green with his second and walked away with a bogey.
Meanwhile, Fasth came up short, left of the putting surface. Not to worry though as he chipped in for birdie. He stood at minus-4, one clear of Murphy.
Murphy found a greenside bunker at the 12th and could not save par from there. He dropped to minus-2, two behind his playing partner Fasth.
Trouble was looming for Fasth though. He lost his drive into the trees right of the fairway at 14 and knocked his approach over the green on 15. He bogeyed each of those holes to drop to 2 under, one behind Murphy.
Murphy moved back to minus-3 as he converted an 8-foot birdie putt on 14. Fasth got back into a share of the lead when he poured in a 20-footer for birdie on 17. They both parred the last to share the lead for the third straight round.
'Generally I'm pretty happy with the round. I left a few shots out there, but its going to happen on a course like that,' Murphy admitted. 'I have played seven through nine well all week and didn't play them too well today. I'm still happy to be leading though.'
For the Irishman Murphy, he remains in solid position to retain his tour card for next year. He currently stands 118th in the Order of Merit, which is the final spot for players to retain their cards for next year. A big pay day here would secure him his card for next season.
Gregory Havret fired a 4-under 66 Saturday to climb into a share of eighth place at 1-over-par 211. He was joined there by Phillip Archer and Soren Kjeldsen.
Among the six players tied for 11th are Carlos Rodiles and Robert Karlsson. Rodiles had the low round of the day with his 6-under 64. It will not count as a course record because players are able to place their balls in the fairway.
Karlsson, who shot even-par 70 Saturday, is fourth in the Order of Merit race. He could overtake Paul Casey for the top spot with solid finishes this week and next. A win this week would get him into second place behind Casey. Padraig Harrington, No. 2 in the Order of Merit, missed the cut here this week, while No. 3 David Howell took the week off.