Bayern Munich had already secured top spot in Group B with Spurs guaranteed second ahead of kick-off.
The home side had chances to kill the game off but failed to take them, before Loic Remy pulled one back for Lille which silenced the crowd in London.
The Arsenal boss hopes a win at Standard Liege will build confidence and get fans on side.
Having already qualified for the knockout stages, manager Ole Gunnar Solskjaer opted for youth but was left disappointed.
Harry Kane becomes the fastest player to score 20 goals in Champions League history.