The England striker moved onto 98 goals, surpassing the previous mark of 97 set by Teddy Sheringham, thanks to two tap-ins at Wembley Stadium.
Chelsea, another team fighting for a finish in the Champions League qualification positions, had it much easier in a 5-0 victory over a depleted Stoke side at Stamford Bridge.
Moyes' side got off to a terrible start with Angelo Ogbonna failing to clear Jamie Vardy's cross, allowing Marc Albrighton to slide in for Leicester's opener in the eighth minute.
Xherdan Shaqiri levelled once for Stoke, and substitute Peter Crouch equalized the second time with a header in the 73rd minute.
The recalled Demarai Gray starred as he helped create the opener for Jamie Vardy before adding a fortuitous second himself against struggling Everton on Sunday.
There were wins for Manchester United, Leicester and Bournemouth, but fellow Premier League team Crystal Palace came unstuck in losing 4-1 at second-tier Bristol City.
Coutinho has been building up to full fitness after missing the start of the season while he attempted to force a move to Barcelona.
Leicester couldn't handle Coutinho's passing and movement in the first half, and only came into the game when he was off the field.
Four of City's goals against Liverpool came after Sadio Mane was sent off for a high challenge that forced goalkeeper Ederson off injured in Manchester.
Jamie Vardy smashed one back for the hosts from the penalty spot, but Antonio Conte's team held on for the three points.
Arsene Wenger's side was trailing 3-2 with seven minutes remaining but Ramsey grabbed the equalizer and Giroud headed in an 85th-minute winner.
Chelsea ultimately staved off the ignominy of being consigned to the lower divisions, finishing the 2015-16 season at a lowly 10th place.
London: Leicester City secured an incredible first English Premier League title without playing on Monday after second-place Tottenham drew 2-2 at Chelsea in one of the biggest-ever triumphs by an underdog in sports. Leicester City
London: English football club Arsenal fears midfielder Aaron Ramsey is set to be out for at least four weeks after suffering a hamstring injury against Leicester City on Wednesday.Ramsey lasted only nine minutes before succumbing
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