Cape Town - Arsenal failed to finish in the Premier League's top-four for the first time under Arsene Wenger, despite beating Everton 3-1 with only 10 men at the Emirates on Sunday.
The hosts made a bright start when Sanchez used his vision to unleash Bellerin in as early as the second minute. The Spaniard crossed in for Danny Welbeck, who was charging towards goal, but he missed the target.
Wenger's side didn't have to wait long for another opportunity. Mesut Ozil played in Welbeck, who completely missed an open goal from very close range, but thankfully for him, Bellerin was on hand to finish the move.
Their high spirits almost fell five minutes later, though, when Enner Valencia was in space down the right-hand side of the field and Koscielny flew in with a sliding challenge, sending the winger tumbling to receive a straight red card.
But the sending-off did little to dampen their mood as Sanchez scored his 24th league goal of the season, the Chilean on hand to convert Welbeck's cross from close range.
The Toffees had a number of chances leading up to half-time. These included a goalmouth scramble, a long-range effort from Idrissa Gueye and a looping header from Lukaku; all of which were dealt with by Petr Cech.
The first five minutes of the second half saw Ronald Koeman's men on the back foot when Joel Robles was required to make two consecutive saves. Welbeck fired goalward, which the keeper blocked towards Ozil, who also forced a save out of the Spaniard.
Everton were awarded a penalty in the 57th minute when Nacho Monreal got a hand to the ball. Lukaku stepped up and took a perfect penalty to reduce the deficit despite Cech diving correctly to his right.
Valencia thought that he had scored the equaliser in the 67th minute. Mason Holgate whipped in a cross, which Valencia headed into the goal, but it was disallowed as the striker climbed onto the defender's back illegally.
Arsenal had an opportunity score a third on 75 minutes when Ramsey played a perfectly weighted through ball to Ozil, but Robles was again solid in goal.
With just over three minutes left Alex Iwobi broke the backline of Everton and found himself one-on-one with the keeper, but his attempt was blocked.
They weren't finished there, though as Ozil dribbled towards the box with some wonderful skill and set the ball to Ramsey, who curled the ball into the top-right corner for a spectacular third goal in added time.