Manchester United booked their spot in the top four after booting Leicester City out of the coveted UEFA Champions League places with a 2-0 win.
Leicester sat deep in their own half in the early stages and although they were threatening on counter-attacks, they nearly paid the price for it in the 32nd minute after Bruno Fernandes found the back of the net for United from Paul Pogba's through ball but the goal was disallowed for offside.
Solskjaer's troops almost went to the break with a 1-0 lead after a corner dropped to Rashford at the back post but the English forward fired a strike at close range that was parried away by shot-stopper Kasper Schmeichel.
Jamie Vardy came the closest to finding the breakthrough on the hour-mark after his header inside the box looped over goalkeeper David de Gea towards the back post but it clipped the top corner.
United had a glorious chance to break the deadlock in the last 20 minutes from the penalty spot after Jonny Evans brought down Anthony Martial inside the box, with the Video Assistant Referee confirming the penalty.
Fernandes stepped up to take the spot-kick and he finally gave the Red Devils a 1-0 lead, sending Schmeichel the wrong way.
Wes Morgan had a brilliant chance to level matters for the hosts 11 minutes before time after he received a cross unmarked inside the box but he fluffed his volley with a left-footed strike.
Leicester were pressurising United in the closing minutes, desperate to get back into the game but their chances disappeared in second-half stoppage time as they were reduced to ten men after Evans got a straight red card for a studs up challenge on Scott McTominay.
United went 2-0 up in virtually the last action of the game after late substitute Jesse Lingard pressed Schmeichel, took the ball away and slotted into a gaping net.
Chelsea 2-0 Wolverhampton Wanderers
Chelsea followed suit with a 2-0 victory over Wolverhampton Wanderers in the last matchday of the English Premier League on Sunday afternoon.
The Blues boss Frank Lampard made a big call by dropping goalkeeper Kepa Arrizabalaga after conceding five goals against Liverpool, as Willy Caballero was recalled in goal.
Caballero and the Wolves goalkeeper Rui Patricio both did not have much to do in the first half-an-hour, as the sides did not have any shots on target, cancelling each other out in a cagey affair.
However, despite a lack of tempo Jonny almost broke the deadlock for Wolves three minutes before the break with a cross-cum-shot that was looping over Caballero but the Argentine made a crucial stop back-pedalling back to his goals.
Mount eventually handed Lampard's charges the lead in first-half stoppage time with a stunning free-kick from the edge of the box that curled into the bottom corner.
Chelsea went to the break 2-0 up after scoring deep in added time of the first stanza, as Mount won possession high up the pitch and played in Giroud, who had to round Patricio and Conor Coady before slamming a strike into an empy net.
Wolves were revived early in the second stanza after Nuno Espirito Santo introduced speedy forward Adama Traore in place of Pedro Neto before Diogo Jota had a chance to pull a goal back on the hour-mark with a strike from the edge of the box that was smothered by Caballero.
Chelsea were had the game by the scruff of the neck in the latter stages, threatening to put the game to bed as Lampard refreshed his side with a double substitution in the last 12 minutes, as Callum Hudson-Odoi came on for Pulisic and Tammy Abraham replaced Giroud.
Abraham almost added a third for the Blues in stoppage time on a counter-attack but his shot was saved by Patricio as the hosts had done enough to clinch a fourth spot while Wolves finished in seventh place.