The Red Devils made a worrying start to the Premier League clash after Germany international Antonio Rudiger broke the deadlock in the first half at Stamford Bridge.
However, former AS Monaco forward Anthony Martial levelled with an instinctive finish before the 22-year-old handed the visitors a slender lead with a goal on the edge of the area.
Manchester United had the chance to move to within four points of Chelsea FC but Maurizio Sarri’s men netted a late equaliser in added time at the end of the second half.
The Red Devils appear to be making progress under Jose Mourinho after collecting four points from their last two Premier League outings.
Manchester United have more points in the Champions League than any of their Premier League rivals after securing four points from their opening two games against Young Boys and Valencia.
Shaw, who signed a new long-term deal with Manchester United last week, is hoping that Mourinho’s side can build on their performance at Stamford Bridge when Juventus make the trip to Old Trafford on Tuesday night.
“Chelsea had a few set pieces that caused us problems, but I think in open play, they didn’t test us as much as they have tested other teams this season with the quality they have,” Shaw told MUTV.
“We came in at half-time 1-0 down, but we knew we didn’t have to change much, maybe push a little bit more because I thought when we had the ball we were playing football, but the final ball in the first half was lacking that cutting edge.
“In the second half we were better, more like ourselves and we were playing football and defending well as a team and our counter-attack was good for the goals. Martial had two very good finishes, and we should have won the game,” he added.
“We now have a very important game on Tuesday that we need to focus on and we need to win because it is important to win our home games in the Champions League. So we need to take the positives from the second half against Chelsea and move forward.”
Manchester United eased to a resounding 3-0 win over Young Boys in their opening Champions League fixture of the season to secure a vital three points.
The Red Devils were held to a frustrating goalless draw with La Liga side Valencia as Manchester United struggled to produce an incisive performance.
Juventus are sitting at the top of the group following successive victories over Valencia and Young Boys ahead of their trip to Old Trafford.
Manchester United legend Cristiano Ronaldo is set to make a return to Old Trafford following six successful seasons under Sir Alex Ferguson.
Ronaldo helped the Red Devils to win the Champions League title in 2008 after a penalty shootout victory over Chelsea FC in Moscow.
25/2/2003 – Juventus – Manchester United FC 0-3 (Turin) Giggs 15, 41, Van Nistelrooy 63
19/02/2003 – Manchester United FC – Juventus 2-1 (Manchester) Brown 4, Van Nistelrooy 85; Nedvěd 90+2
21/04/1999 – Juventus – Manchester United FC 2-3 agg: 3-4 (Turin) F. Inzaghi 6, 11; Keane 24, Yorke 34, Cole 83
7/4/1999 – Manchester United FC – Juventus 1-1 (Manchester) Giggs 90+2; Conte 25
10/12/1997 – Juventus – Manchester United FC 1-0 (Turin) F. Inzaghi 84
01/10/1997 – Manchester United FC – Juventus 3-2 (Manchester) Sheringham 37, Scholes 69, Giggs 89; Del Piero 1, Zidane 90
20/11/1996 – Manchester United FC – Juventus 0-1 (Manchester) Del Piero 36 (P)
11/09/1996 – Juventus – Manchester United FC – 1-0 (Turin) Bokšić 34
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