Albian Ajeti came off the bench to save Celtic from dropping points for the second time in their opening three league games as the Scottish champions edged out Dundee United 1-0 at Tannadice on Saturday.
Celtic remain six points behind Rangers, who were 2-0 winners over Kilmarnock, but with two games in hand after having their last two league matches postponed due to left-back Boli Bolingoli's breach of quarantine regulations after a trip to Spain.
Ajeti failed to score at all for West Ham after an £8 million ($10.5 million) move from Basel a year ago, but the Swiss international took just eight minutes into his Scottish Premiership debut to finally turn Celtic's dominance into a winning goal.
Neil Lennon's men were wasteful in front of goal as Odsonne Edouard hit the post early on before Benjamin Siegrist made a string of fine saves from the French striker and Ryan Christie.
"We should have won the game by a lot more. We totally dominated the game in terms of attacking play and possession, we were just a bit loose with our finishing," said Lennon.
However, with their 22nd shot on goal, the visitors made the breakthrough seven minutes from time when for once Siegrist could only punch a Christie shot from distance back into the danger area and Ajeti was alive to sweep the ball home from close range.
"I think it's a big goal for him and a big goal for us," added Lennon. "Already he's made a big contribution and hopefully once he's fully match fit we'll see the best of him."
Rangers are yet to concede in their opening five games of the season as they aim to stop Celtic's quest for a record 10 titles in a row.
But it was Steven Gerrard's selection at the other end of the field at Ibrox that sparked further speculation over Alfredo Morelos's future as the Colombian was left out of the former Liverpool captain's 20-man squad.
"You pick your team, you pick your squad on what you see on a daily basis and every single person in the squad deserved to be there," said Gerrard, who said earlier this week Morelos's head had been turned by rumours of a move to Lille.
Kemar Roofe and Cedric Itten were handed their first starts in the absence of Morelos and it was Roofe who made the breakthrough after a frustrating first 45 minutes for the hosts.
The former Leeds forward turned home Borna Barisic's cross five minutes into the second half.
Gerrard said on Friday a bid from Leeds for Ryan Kent had been turned down, but unlike Morelos, the former Liverpool winger is showing no sign of being distracted by transfer talk and secured the three points from James Tavernier's cross for his third goal of the season.
At the other end of the table, Hamilton got their first win of the season with a 1-0 success at local rivals Motherwell thanks to Hakeem Odoffin's late header.
Ross County remain third after coming from behind to earn a 1-1 draw at St Mirren.