De Gea had been in the spotlight for his poor performances as Solskjaer's men had lost heavily to Barcelona, Everton and Manchester City in their previous three games.
And after a bright start from the hosts was rewarded by Juan Mata's 11th-minute opener, De Gea cost the Red Devils again when he spilled Antonio Rudiger's long-range effort to allow Marcos Alonso to equalise just before half-time.
Yet, as De Gea headed for the dressing room at the break, his name was chanted by the home support in recognition of his brilliant performances over a number of seasons prior to his recent loss of form.
"David has been unbelievable for this club. That just says everything about the supporters," said Solskjaer.
"No chance anyone can blame him for losing many points for us."
De Gea's future at United is uncertain, though, as he has just a year left on his contract and Solskjaer admitted he will talk with his goalkeeper this week to see if it is best to withdraw him from the limelight for the rest of the season.
"I think David is one that would like to play games. I'll chat with him this week and I'm sure he'll respond in the right way."
- Chelsea in command -
A point halts a run of seven defeats in nine games for United, but does little for their chances of overhauling Chelsea into the top four.
Maurizio Sarri's side remain three points ahead of United and have a superior goal difference with just two games of the season remaining.
Arsenal's defeat by Leicester earlier on Sunday means Chelsea also open up a two-point gap on the Gunners.
"We need two wins, with two wins we are sure, with four points it depends on goal difference," said Sarri, who defended his record during an at times difficult first season at Stamford Bridge.
"We played the final in the League Cup, we are fighting for the top four, we are in the semi-finals of the Europa League.
"I think we have done a good season. Now we need to take two very important targets."
Solskjaer made five changes from the side beaten comfortably by Manchester City in midweek and got an instant response with United's first goal from open play in two minutes short of nine hours.
Romelu Lukaku's lofted pass picked out the run of Luke Shaw from left-back and his cut-back found Mata to roll into an unguarded net against his former side.
Chelsea were on course for a clean sweep of five defeats away to top-six opposition this season.
However, they were given a lifeline two minutes before half-time as De Gea's miserable run continued.
That blow appeared to drain United's belief after the break for what Solskjaer had termed a "must-win" for his side.
"I don't think you can give up until it is theoretically over," said Solskjaer on his side's chances of making the top four.
"It's a big mountain to climb. We need to focus on our two games, win games again and see what the other games throw up."
Both sides were hit by defensive injuries in a disjointed second-half as Rudiger and Bailly hobbled off.
Bailly's replacement Marcos Rojo came closest to grabbing a winner for United when his header was cleared off the line by Pedro Rodriguez in stoppage time.
And Chelsea could have grabbed all three points with virtually the last kick of the game when the ineffective Gonzalo Higuain shot straight at De Gea from a narrow angle.