The Hoops began life at the top of the Clydesdale Bank Premier League in convincing fashion as Mulgrew scored one and helped set up the other two goals at East End Park. The former Aberdeen defender capped a fine display by curling home a 30-yard free-kick.
Lennon said: "He's having a great time of it at the minute. I thought he had a great game on Wednesday (against Rangers). His delivery wasn't quite there for a couple of weeks but now he's found his range. The ball for the second goal was sublime and his free-kick was a piece of individual brilliance."
Bottom club Dunfermline have not won at home all season but Lennon had warned against complacency after a weekend which saw Manchester United and Manchester City suffer shock defeats.
"It's hard winning games," Lennon said. "I said to the players that Dunfermline would make life very difficult for us, which they did.
"You only have to look at some of the results in the (Barclays) Premier League this weekend to see what can happen when you take your eye off the ball. But thankfully we got the job done.
"We could have added more goals towards the end but I'm very, very happy with the performance."
Dunfermline manager Jim McIntyre was happy with his side's work rate but disappointed with some of the goals conceded.
McIntyre said: "It was a great finish from (Anthony) Stokes, there's not a lot you can do about that, he has stuck it in the postage stamp.
"But the other two goals were definitely preventable. The second goal we have lost a man in the box and he's got a free run, and the third goal the goalkeeper should have saved. In the first half we had a wee chance to get the third goal with (Paul) Burns. That's a big moment - you need to take them against the Old Firm."