Gordon kept eight successive clean sheets between December and February, which almost became nine until a 72nd minute St Johnstone goal ended the run on 14 February 2015. An outstanding double save near the end ensured a 2–1 win for Celtic. Gordon endured a torrid night in the Europa League against Inter Milan on 19 February 2015, being culpable for Inter's opening goal and at fault for their third before half-time; Celtic rallied to square the match at 3–3 with an injury time goal, and Gordon went some way to redeeming his earlier errors by pulling off an outstanding save from Xherdan Shaqiri's free kick in what proved to be the last action of the game. In the second leg at San Siro, Gordon kept 10-man Celtic in the tie with a number of excellent saves until the 87th minute when Fredy Guarín scored with a powerful strike to give Inter a 1–0 win on the night and a 4–3 aggregate victory. One of Gordon's saves away against Inter was compared by Scottish newspaper, the Daily Record, to the Gordon Banks save against Pele in 1970.
Gordon won his first silverware with Celtic after they defeated Dundee United 2–0 in the Scottish League Cup Final on 15 March 2015. He kept a clean sheet in every round of the competition, which included playing in the first Old Firm game for three years. According to The Scotsman in March 2015, Gordon was attracting the attention of Chelsea. He made his 50th Celtic appearance in a 2–1 league win at Dundee on 22 April 2015. A day after shutting out Dundee in a 5–0 home win, Celtic were declared league champions after an Aberdeen defeat at Dundee United on 2 May 2015. It was his first league title as a player. Gordon played 52 times and kept 28 clean sheets in his first season at Celtic. Gordon was voted Player of the Year by the Scottish Football Writers' Association, and was selected in the PFA Scotland Premiership Team of the Year. He was omitted from the PFA Players' shortlist, despite having been touted amongst the favourites to win it.
For the 2015–16 season, Gordon was assigned the number 1 shirt to wear replacing the number 26 shirt he was handed when he had signed. In July 2015, Gordon signed a new contract with Celtic until 2018. His first match of season 2015–16 was the 2–0 UEFA Champions League second qualifying round first leg win at home to Stjarnan on 15 July 2015. Celtic progressed 6–1 on aggregate. The 1–0 win against Qarabağ, in the third qualifying round first leg at home, was his 30th clean sheet in 55 appearances for Celtic. Celtic advanced 1–0 on aggregate. Gordon also started the league campaign with consecutive clean sheets against Ross County and Partick Thistle. However, Gordon's form at this time was generally poorer than the previous season, in part due to Celtic struggling to find a settled defensive line-up following the departures of centre-halves Virgil van Dijk and Jason Denayer.
Celtic again won domestic trebles in 2017–18 and in 2018–19. This meant they had won all three competitions for three consecutive years, a feat known as the "treble treble". Gordon played regularly during that time but occasional high profile mistakes, such as in the UEFA Europa League against Fenerbahçe in 2015 and Red Bull Salzburg in 2018, posed doubts as to his relaibility; he lost his place to Fraser Forster (returning from Southampton on loan) in 2019–20, and made only six first team appearances during that season. He decided to leave Celtic in June 2020, despite having been offered a new contract, as he wanted to play more regularly.