Monday saw second-placed Spurs lose 2-1 at home to Southampton before the Gunners, who are third, secured a 2-2 draw at Manchester City.
It left Arsenal two points behind Tottenham with both sides having one game to go - a home clash with rock-bottom Aston Villa for the former and 18th-placed Newcastle away for Mauricio Pochettino's men.
Arsenal midfielder Wilshere told the London Evening Standard: "We all knew the Spurs result (before the City game) and of course that motivates us.
"We want to finish above them and show we are the best team in north London.
"We expect to win our home games so hopefully Newcastle can do us a favour.
"I saw about 10 minutes of the first half and when I turned it off it was 1-1.
"Obviously, then you are checking your phone in the dressing room. When we saw it finished 2-1 it gave us a real lift."
Arsenal twice came from a goal down to level against City, Olivier Giroud swiftly replying to Sergio Aguero's early opener and Alexis Sanchez then striking with just over 20 minutes left to cancel out Kevin De Bruyne's effort.
Champions League football is not yet certain for Arsene Wenger's men, but the result meant they remained in control of their own destiny in terms of finishing at least third and thus securing automatic qualification for next season's group stages.
They are three points clear of fourth-placed City and five better off than Manchester United, who are fifth having played a game less than the sides above them.
"The main thing was not to lose so it was still in our hands. After going 1-0 down and then 2-1 down, overall you would take a draw," Wilshere said.
"It is a tough place to come and we showed good character - that's what we can take from the game."
Wilshere, who came on in the 24th minute to replace the injured Danny Welbeck, was making only his second appearance of the season as he continues his comeback from a lengthy absence.
And the 24-year-old, hoping to feature for England this summer at the European Championship, was delighted to get a decent run-out in a match of considerable intensity following his injury problems.
"As a player, these are the games you want to be involved in," said Wilshere, quoted by Arsenal's official website.
"When the game started, I was watching it thinking it was high intensity and this is what I need.
"I got to come on and get 70-odd minutes under my belt, so I'm happy."
For City, it was a hugely frustrating way to mark what was boss Manuel Pellegrini's last home game in charge before he is replaced by Pep Guardiola this summer.
The Blues are at Swansea in next week's final round of fixtures - and know by then they could have been pushed into fifth.
They will be watching with interest on Wednesday when West Ham host Manchester United, who then entertain Bournemouth on Monday.