Louis Saha, Victor Anichebe and James Vaughan were all struck down by injury.
Anichebe curled an effort too close to the prone Scott Carson, and was denied by the home keeper's outstretched foot.
Victor Anichebe had two headers off target before Everton took the lead with a freak goal eight minutes before the interval.
Moyes did dispute Cahill's 10th-minute headed opener which came from a free-kick where Victor Anichebe was adjudged to have fouled Paul Robinson.
BBC: Loss at Bolton is one of worst Everton displays - Moyes
Papiss Cisse had given Newcastle a second minute lead before Leighton Baines' stunning free-kick and Victor Anichebe's strike won it for Everton.
"I've never felt as much pain in my life, " Anichebe told evertonTV.
William Blything shouted abuse at Queens Park Rangers' Korean captain Park Ji-Sung and Everton forward Victor Anichebe at Loftus Road on 21 October.
Coyle said that he wished Zat Knight a speedy recovery after the defender left the field with knee ligament damage after a challenge with Anichebe.
BBC: Loss at Bolton is one of worst Everton displays - Moyes
Despite some probings by Anichebe, the Everton midfield could not find the crucial pass to their forwards, with the Nigerian and substitute Jermaine Beckford left frustrated.
Moyes opted to replace Beckford with Anichebe and the move almost paid immediate dividends as the Nigerian stretched but narrowly failed to convert another Bilyaletdinov centre.
Everton lost Victor Anichebe to a second booking on the hour and City pulled a goal back when Yaya Toure's shot deflected in off Phil Jagielka.
Sole striker Anichebe played his part, flicking Baines' pass into the path of Seamus Coleman before Cahill promptly headed the Irishman's incisive cross beyond Joe Hart.
Midfielders Steven Pienaar (hamstring) and Leon Osman (stomach strain) have both missed recent games, while Anichebe was injured by a horror tackle from Kevin Nolan against Newcastle.
City were stunned and almost fell further behind when Beckford stole in on Victor Anichebe's cross only to be denied by Hart's crucial block at the near post.
Moyes is also worried over the fitness of Victor Anichebe, who was replaced shortly before half-time after a heavy tackle from Kevin Nolan, which saw the Newcastle midfielder receive a straight red card.
Anichebe, in only his third league game this season, set off on a storming run down the left, crossing to Cahill who laid the ball neatly off to defender Baines who curled a shot beyond Hart.
Eager to put some distance between themselves and the bottom of the table, Everton pressed for a second and while defender Pablo Zabaleta was off the pitch with a head injury, Anichebe and Cahill combined again to create a second goal on 19 minutes.
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