The Scot got the better of Israel’s Lonah Chemtai Salpeter towards the end but could not catch Can, crossing the line 8.48 seconds behind in a time of 30mins 41.05secs.
McColgan was seeking further glory after securing a first major title with her gold earlier this month at the Commonwealth Games in Birmingham, where she also claimed 5,000m silver.
The 31-year-old told BBC Sport: “I’m pleased. I’m obviously disappointed – I would have loved to have become European champion tonight. But tired, tired legs – I definitely felt that today.
“It’s been probably one of the craziest weeks of my life at the Commonwealth Games, such a big high of emotions and not a lot of sleep. So I found it tough to sort of get myself up again today, but I can’t complain. I did a lot of work tonight and came away with a silver medal.”
When asked about doing the 5,000m in Munich, McColgan said: “I think it’s going to be tough, two days’ recovery to try to do it all again.
“But this will be my last track race this season. We’ll have a think about it tonight. I’m sure I’ll be back, I love racing. It’s been a great season so far and to add another medal to that, I’m more than proud.”
Fellow Britons Samantha Harrison and Jessica Judd finished sixth and 10th respectively.
In the shot put, Scott Lincoln came 10th and Sophie McKinna was 12th, and the best British finishers in the marathons earlier in the day were Philip Sesemann (17th) and Alice Wright (22nd).
Day one of the athletics programme also saw team captain Ashleigh Nelson advance into the women’s 100m semi-finals, coming third in her heat in a time of 11.41s.
Nelson is competing in the event as a replacement for Asha Philip, who was forced to withdraw following a positive coronavirus test.
— British Athletics (@BritAthletics) August 15, 2022
On the penultimate day of the track cycling programme, GB were unable to add to their medal haul as Pfeiffer Georgi and Oliver Wood missed out on omnium podium places.
Georgi end up seventh in the women’s event while Wood came eighth in the men’s.
Kian Emadi had earlier failed to make the final of the one kilometre time trial, finishing 12th in qualifying.
Britain have so far accrued five medals in the velodrome, including Georgi securing elimination race silver and Wood contributing to a team pursuit bronze.
Meanwhile, it was announced that Jack Carlin, who on Sunday crashed in the individual sprint semi-finals before going on to claim silver, was flying home and would not compete in the keirin on Tuesday.