The Champions almost went behind in what was a breathless opening few minutes in the game at Drumahoe when Stephen O'Flynn struck the post, but it was the Reds' new signing David McDaid who made the breakthrough just a minute later when converted Marty Donnelly's cross from close range.
However the top flight newcomers appeared keen to make a fight of it and got themselves back on terms with a deflected effort from Declan McKeever in the ninth minute.
Cliftonville were back in front in the 16th minute when Chris Curran found himself one-on-one with keeper Michael Doherty and he kept his composure to stroke the ball home.
Stephen Curry made it 2-2 in the 28th minute when he pounced on O'Flynn's pass before finishing high into the roof of the net, and as the game ebbed and flowed it appeared that a draw would be a fair result.
Cliftonville though had other ideas, and a smart ball from McDaid found Garrett in space, and the midfielder steered the ball past Doherty to claim all three points for the visitors.
Ballymena United remain joint-leaders alongside the Reds thanks to a hard-fought 3-2 victory over Portadown at the Ballymena Showgrounds.
Kyle McVey opened the scoring for the hosts when he headed in from a corner on the half-hour, but Robbie Garrett equalised in the 66th minute with a low shot past Dwayne Nelson.
Sky Blues captain Allan Jenkins restored his side's lead when he played a neat one-two with Neal Gawley before firing into the visitors net, and Darren Boyce made it 3-1 when he pounced on a loose pass before running through to score past David Miskelly.
Darren Murray netted a late penalty for the Ports, but it proved a mere consolation as Ballymena held on for the victory.
In the game at the Coleraine Showgrounds Linfield defender Matthew Clarke bagged a brace to help his side to a 2-1 victory over hosts Coleraine.
His first was excellently worked, with Keith Miller teeing up the full-back with a cheeky back-heel in the 54th minute, allowing Clarke to dispatch the ball into the roof of the net.
Coleraine responded with a superb curling free kick from Neil McCaffrey which left Blues stopper Jonny Tuffey with no chance, but Clarke restored his sides lead in the 62nd minute when he steered a free kick of his own past Eugene Ferry and in off the post.
The Bannsiders ended the game with ten men when Joe McNeill was given a straight red card after appearing to aim a headbutt at Linfield's Aaron Burns in the 68th minute.
Elsewhere Crusaders came from behind to pick up their first win of the new campaign with a 2-1 victory over Ballinamallard at Seaview.
Johnny Laverty gave the visitors the lead with a brilliant long-range effort, only for Paul Heatley to score in equally impressive fashion with a spectacular volley in the 37th minute.
Josh Robinson put the Crues in front with a 64th minute strike, and netted his side's third when he headed in at the back post three minutes from time to seal the points for Stephen Baxter's side.
Glentoran also picked up all three points for the first time this season with a 3-1 victory over Warrenpoint Town.
The Co Down side struck first when Stephen Hughes headed in at the near post after 21 minutes, but the Glens responded well with Curtis Allen restoring parity with a lobbed effort ten minutes later.
Fra McCaffery added a second for the hosts when he finished from close-range after being played in by Allen, who added his second of the Afternoon in the 89th minute to make sure of the victory.
Finally in the game at Mourneview Park Glenavon suffered a 2-0 defeat at home to Dungannon Swifts.
Two goals in three minutes from Gary Liggett and Ryan Harpur put the visitors in control by the 12th minute, and the Swifts never looked like surrendering their lead as they picked up a victorytt that sees them move up to fifth in the league table.
Saturday's Irish Premiership results:
Ballymena United 3 - 2 Portadown
Coleraine 1 - 2 Linfield
Crusaders 3 - 1 Ballinamallard Utd
Glenavon 0 - 2 Dungannon Swifts
Glentoran 3 - 1 Warrenpoint Town