Both sides finished with ten men in what was a bad-tempered affair, and both squandered genuine changes to take all three points.
Mark McAllister struck the post for the blues in the 29th minute, but David Jeffrey's side suffered a blow two minutes later when Aaron Burns was shown a red card after apparently lashing out at Ryan O'Neill.
Jason Hill was sent off for a second bookable offence in the 74th minute, and Richard Clarke had the Glens' best chance five minute later, but his wickedly curving strike was turned away by Jonathan Tuffey.
The result means the Blues languish at the bottom of the table with just two points from their opening four games, while Glentoran move to fifth in the table.
In Friday's Premiership games, champions Cliftonville lost 2-1 to Portadown.
The Reds claimed the first goal at Solitude with a deflected effort from Marc Smyth in the 69th minute, but Chris Casement equalised for the visitors with a superb free kick.
The game looked to be heading for a draw until Gary Twigg fired into the roof of the net late on to seal the win and bring to an end a 24-game undefeated streak for Tommy Breslin's side.
Elsewhere on Friday newly-promoted Ards picked up their first victory of the season with a 2-0 win over Ballymena United.
Ben Roy opened the scoring for the visitors with a goal in the 32nd minute, with William Faulkner adding a second from an outrageous angle 11 minutes from time to seal the points for Niall Currie's side.
In the game at the Coleraine Showgrounds a brace from Timmy Adamson saw Crusaders draw 2-2 with Coleraine
Gary Browne had nodded the hosts in front in the 57th minute before Adamson weighed in with the equaliser in the 69th minute as he capitalised on a loose ball after Wayne Drummond had spilled Gary McCutcheon's free kick.
The Bannsiders went down to ten men after Aaron Canning's frank exchange with the referee saw him earn a red card, but Ruairi Harkin's brilliant long-range free kick gave Oran Kearney's side the lead in injury time.
Coleraine might have thought the points were safe, but two minutes later Adamson rose unmarked to head home the equaliser and condemn the Bannsiders to a fourth straight draw.
It was also a draw at Stangmore Park, with Dungannon Swifts and Glenavon sharing six goals between them in a 3-3 draw.
Dungannon were in the driving seat after a William Murphy own-goal and a header from Richard Copeland put them 2-0 up with half an hour gone.
However Glenavon recovered well and Murphy made amends for his earlier mistake by heading in a long throw-in from Gareth McKeown in the 60th minute, before Guy Bates added the equaliser three minutes later.
The Lurgan Blues took the lead in the 70th minute when Mark Patton found the net, but Dungannon secured the point with 15 minutes left on the clock when
Darren Boyce converted from the spot after Shane McCabe had fouled Ryan Harpur in the area.
Finally on Friday night Warrenpoint Town suffered a 3-0 defeat at the hands of Ballinamallard Utd, a result which moved the Mallards up to third in the table.
Jason McCartney weighed in with two goals in the dying moments of the first half to put his side in the driving seat.
His first came from the spot after a foul on Mark Stafford, and McCartney added another deep into first half injury-time when he was quickest to react after Nathan Cashel's header hit the bar.
The midfielder then turned provider as he raced away from the Warrenpoint defence before teeing up Richard Leckey to score a simple tap-in.
Danske Bank Premiership results:
Saturday 31 August:
Linfield 0-0 Glentoran
Friday 30 August:
Ards 2-0 Ballymena Utd
Ards: Ben Roy (32), Willie Faulkner (79)
Cliftonville 1-2 Portadown
Cliftonville: Marc Smyth (68)
Portadown: Chris Casement (83), Gary Twigg (90)
Coleraine 2-2 Crusaders
Coleraine: Gary Browne (57), Ruairi Harkin (90+1)
Crusaders: Timmy Adamson (69, 90+3)
Dungannon Swifts 3-3 Glenavon
Dungannon: Richard Copeland (30), Darren Boyce (75)
Glenavon: William Murphy (O.G 14), William Murphy (60), Guy Bates (63), Mark Patton (68)
Ballinamallard Utd 3-0 Warrenpoint Town
Ballinamallard: Jason McCartney (40, 45+5), Richard Lecky (83)