McTominay, captaining United at Old Trafford, headed home Alex Telles's corner in the fifth minute to secure United's 34th victory from its last 36 third-round ties.
It was the Scotland international's fourth goal of the season in all competitions, one short of his best tally in a single campaign (five goals in 37 appearances last season).
"Scott has had a very good season," Solskjaer told BBC Sport. "He is being more and more dangerous, we know he can be a box-to-box midfielder.
"He should have scored a few more on set-plays before now because he is a monster of a human being. I am very happy he got the goal."
United did not have it all its own way against the Hornets, who attempted 18 shots in the match – the second-most it has had away from home this season.
"The most important thing in the cup is that you go through so that will make everyone happy but the first 15-20 minutes I think were the highlights of our game," Solskjaer said. "I think we played some very good stuff, created chances and then we let them back into the game but we are through.
"I'm delighted we are through. I told them I didn't expect them to start like a house on fire. I expected them to grow into the game. The opposite happened more or less. We started brightly."
McTominay captained United for the first time and led by example, completing 45 of 56 attempted passes, creating one chance and making a joint team-high three clearances.
The 24 year-old told BT Sport: "When the manager told me it was a surreal moment.
"I've been here since I had just turned five, so that's 18 or 19 years associated with the club and it is a huge honour. I love this club and it has been my whole life."
McTominay and goalkeeper Dean Henderson were the only two players to keep their places from the side that lost to Manchester City in the Carabao Cup on Thursday (AEDT) and the midfielder acknowledged a general lack of match fitness may have contributed to United's sluggish second-half display.
"We started really well and then for the first half an hour we were really good, moved the ball really well and there was some good combination play," McTominay added.
"Then we couldn't find it after that as well. The second half I didn't think we played anywhere near our level.
"For the boys coming into the team it is difficult, they haven't played for a long time. The manager said 'get your rhythm' and he probably wasn't expecting the rhythm we got in the first 30 minutes. But then there were some tired minds, but credit to the boys."