Christian Eriksen cancelled out Patrick van Aanholt’s opener for Sunderland before Mousa Dembele edged Tottenham into a second-half lead at White Hart Lane.
The Denmark international scored his second of the afternoon before Harry Kane sealed three points from the penalty spot after Danny Rose was brought down.
Former Manchester United midfielder Hargreaves believes Spurs should target third or fourth place, adding if the north London side are in the mix heading into the final games of the season, a potential title challenge could be on the cards.
“They’d be very happy with a top-four finish,” Hargreaves told BT Sport.
“If they’re in the mix towards the end of the season, they’d give it a go.
“But a third or fourth place would be a great achievement for a young Tottenham side.”
Spurs are four points behind Premier League leaders Arsenal following their 4-1 victory over Sam Allardyce’s Sunderland side at The Lane.
The north London side make the trip to Alan Pardew’s Crystal Palace side in the Premier League next weekend after their FA Cup third-round replay at Leicester City mid-week.
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