Sydneysider Glover, who signed for Boro on a three-year deal in July 2023, kept a clean sheet as Michael Carrick's side won 1-0 to leave Chelsea with plenty to do in the second leg at Stamford Bridge on January 23.
"I know it's a two-legged affair and it's still all to play for, but with all we've had to go through with injuries and setbacks to beat a team with Chelsea's quality is unbelievable," Boro manager Carrick said.
"It was an exceptional effort."
Hayden Hackey, 21, scored the only goal of the game with a silky finish from an Isaiah Jones pass in the 37th minute at the Riverside Stadium.
Chelsea only had themselves to blame against their second-tier opponents, dominating possession but squandering their 18 attempts on goal - with England midfielder Cole Palmer twice failing to hit the target with the net beckoning, and forcing a full-length save from Glover early on.
The 26-year-old Australia under-23 international also saved well from Noni Madueke and Mykhailo Mudryk as Chelsea poured forward looking for an equaliser.
"We had too many chances to score where we didn't score - that is football," Chelsea boss Mauricio Pochettino said.
"It is the first half of the tie. We have another 90 minutes at Stamford Bridge and we have to be positive.
"If we assess the performance, overall we were the better side, we created more chances, we had clear chances. But we didn't score."
Boro, who won this competition in 2004, were missing Socceroos Riley McGree and Sammy Silvera - both with Australia in Qatar as part of Graham Arnold's Asian Cup squad.
The hosts, currently 12th in the Championship, had chances on the counterattack to increase their lead but will head to London believing they can reach Wembley.
Chelsea's players were booed by some of their own fans at the final whistle but will have the chance to make amends in the second leg.
Liverpool host Fulham in the first leg of the other semi-final on Wednesday.
- with Reuters
