Born in Nantwich, Cheshire, Westwood started his career in the academy at Crewe Alexandra. He signed his first professional contract in April 2008, along with Luke Danville, Chris Clements, Josh Thompson, AJ Leitch-Smith and Luke Murphy. He joined hometown club Nantwich Town, playing in the Northern Premier League Premier Division, on loan in 2008.
The following season saw Westwood break into first team appearing 38 times for the club along with six goals – the first coming in a 3–2 away win at Chesterfield. The season, however, did see Westwood receive his first career red card in the 0–1 home defeat to Notts County for squaring up to Luke Rodgers after the pair tussled for the ball, the County striker was sent off for the incident as well. The young midfielder impressed the Crewe management over the season and was rewarded with a new two-year contract in December and his performances over the season were rewarded with him named the club's player of the season.
A further 46 appearances followed in the 2010–11 season along with six goals, including two in the 3–3 home draw to Torquay United. Following a string of impressive performances, Westwood signed a new deal intended to keep him at the club until 2014 midway through the campaign.
With the release of David Artell in the summer of 2012, Westwood was handed the club captaincy of Crewe by manager Steve Davis. The young midfielder made three league appearances before he was linked with a move to the Premier League with Swansea. "It's the first time he'll have had this sort of speculation in his career", Crewe manager Steve Davis commented before adding: "he's got to be professional in how he deals with it".
Despite the speculation of a transfer to Swansea City talks between the two clubs broke down as the South Wales club was unable to match the asking price set by the Railwaymen. With talks of the deal reaching a stumbling block, Aston Villa were quick to capitalise and signed the midfielder on the final day of the summer 2012 transfer window. Westwood signed on a four-year deal with the Birmingham based club for a fee believed to be around £2 million.
He made his debut on 15 September 2012, coming on as a second-half substitute for Stephen Ireland during a 2–0 win at home to Swansea City. On 3 November, Westwood made his first league start for Villa in a 1–0 win away at Sunderland. The young midfielder was highly praised for his role in the win, alongside Barry Bannan. On 15 July 2013 Westwood signed a new four-year contract keeping him at the club until summer 2017. He scored his first goal for the club against local rivals West Bromwich Albion in a 2–2 draw at The Hawthorns. On 29 March 2014, he scored in a 4–1 defeat away to Manchester United at Old Trafford. On 3 May 2014, Westwood scored after 54 seconds against Hull City in a 3–1 home victory that helped ensure Villa avoided relegation.
On 12 August 2015 Westwood signed a new five-year contract.
On 31 January 2017, Westwood signed for Premier League club Burnley on a three-and-a-half-year contract, with the option of a further year.
Westwood operates primarily as a deep-lying playmaker although he is capable playing in a role higher up the midfield as well as right back. After his transfer to Aston Villa, Westwood compared himself to Michael Carrick adding that he is "someone who stays out of the limelight and keeps it simple, retaining the ball and laying it off to team-mates".As of match played 19 September 2017