Already-relegated Hearts beat Hibernian 2-1 in the season's final Edinburgh derby, while Celtic thrashed Inverness Caledonian Thistle 6-0.
Hearts sit at the bottom of the relegation group of the Scottish Premiership and have done so for the entire season after suffering a 15-point deduction in June.
They gave their fans a memorable derby win ahead of their drop to the Scottish Championship though, as a pair of first-half strikes from Callum Paterson settled the clash on Sunday.
The Easter Road crowd were silenced by Paterson's opener, as he gave Hearts the lead against the run of play in the 37th minute after meeting a corner with a looping header.
Paterson struck again four minutes later as the striker nodded Kevin McHattie's in-swinging free-kick past goalkeeper Ben Williams.
That left Hibernian shell-shocked, with Terry Butcher's side hovering nervously just one point above the relegation play-off place.
The hosts did manage to pull one goal back - Ryan McGivern's cross was headed in by Jordon Forster with 21 minutes left – but it was not enough as Hearts held on for a fourth win in five league games.
Celtic won far easier at Parkhead as they extended their unbeaten run to eight with a thumping victory.
The champions – who have lost just one league game all season – found themselves 2-0 up at half-time after Anthony Stokes' first-half double.
Stokes completed his hat-trick from the penalty spot in the 53rd minute, giving him 18 league goals for the season.
January signing Leigh Griffiths grabbed a fourth with 22 minutes to go, tapping in after good work from Stokes, and Nigeria defender Efe Ambrose headed in the fifth 10 minutes later.
Substitute Teemu Pukki completed the win moments later with another goal, as Celtic inflicted the league's heaviest defeat of the season on Inverness.comments