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The former US Open Champion recovered from losing the second hole to take five of the next seven against Rory McIlroy’s conqueror.
Lowry took the 15th after McDowell lost his drive then duffed a chip, but a par at the 16th was good enough to send him through to the quarter0final.
“With the possibility of another 18 holes this afternoon, I was very keen to get the match out of the way as soon as I could, especially when I was three and four up,” said the Northern Irishman.
“But Shane is a gritty competitor, very difficult to get the poker face on and really knuckle down and play Shane, being a good friend like he is.
“When he hit that pin with the flop shot on 13, I was licking my chops there thinking I had a chance to go five up with five to play, and all of a sudden I found myself down 16 and really not wanting to go down much past that.”
Australian Jason Day was next for McDowell after a speedy turnaround, and The Ryder Cup star was expecting a tough battle.
“I enjoy the mano a mano of match play and the idiosyncrasies that different players have,” he added.
“I've had a few players this week make me putt short ones, so I have no problem with that. I think I'd probably prefer to play Jason than Bubba - Bubba would have been playing a different golf course from me.
“At least I can play the same golf course as J.D. and hopefully playing the way I'm playing I have a chance to beat him.”
Ian Poulter remains on course for a second WGC Match Play title in Arizona after reaching the quarter-finals with a win over South African Tim Clark.
The Englishman won 5&3 in the third round and will return to action later on Saturday against Steve Stricker in the last eight.
Organisers are looking to get back on schedule after freak snowstorms delayed the start of the competition.
Third-round results (US unless stated):
Graeme McDowell (NI) bt Shane Lowry (Ire) 3&2
Jason Day (Aus) bt Bubba Watson 4&3
Robert Garrigus bt Fredrik Jacobson (Sweden) 3&1
Matt Kuchar bt Nicolas Colsaerts (Belgium) 4&3
Webb Simpson bt Gonzalo Fernandez-Castano 2 up
Hunter Mahan bt Martin Kaymer (Germany) 5&4
Steve Stricker bt Scott Piercy by 1 hole
Ian Poulter (Eng) bt Tim Clark (South Africa) 5&3