Spurs’ 3-1 win over Manchester City on Sunday put them within a point of fourth placed Chelsea and two behind the Gunners with a game in hand.
But Smith believes Arsenal will finish ahead of its north London rival as a Champions League spot represents a "psychological hurdle" for Andre Villas-Boas' side.
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"They did it last year but obviously didn’t get into the Champions League so can they do it again now? Gareth Bale being back will help them but I fancy Arsenal to beat them to it."
In their pursuit of a top-four place, the 50-year-old admits the Gunners' fortunes often hinge on the fitness of England midfielder Jack Wilshere.
"Jack’s so important to the team," Smith said. "I was surprised to hear Arsene Wenger say he probably brought him back too quickly. Wenger’s normally a stickler for not rushing players and bringing them back when they’re properly match fit.
"When he is in the team, he drags them forward. Maybe he does too much, he’s throwing himself all over the place. With Jack being Arsenal’s most important player – I think he’ll be the captain of the team in the not too distant future – they’ve got to look after him."
Having scored the opener in Arsenal’s famous win at Liverpool that sealed the first division title in 1989, Smith admits the Gunners' current plight is difficult to swallow – though he is hopeful of a bright future for the club.
"We hadn’t won the league for 18 years so it was a momentous night for the club and for everybody involved," he explained. "Now, the playing field has changed. With Manchester City and Chelsea coming in and pumping in millions, Arsenal haven’t been able to compete.
"But hopefully in the next few years the financial regulations will work in our favor and the Gunners will be able to compete with the big boys again."
Alan Smith was the keynote speaker at this year’s Isokinetic Football Medicine Conference. The conference is the largest of its kind, attracting delegates from 75 different countries. The conference brings together leaders of football medicine to discuss key issues within the sport and how best to combat injuries.
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